| L(s) = 1 | + 3·4-s + 4·11-s + 3·16-s + 6·19-s − 25-s + 16·29-s − 10·41-s + 12·44-s + 14·49-s − 8·59-s − 28·61-s − 2·64-s − 44·71-s + 18·76-s − 36·79-s − 6·89-s − 3·100-s − 4·101-s + 8·109-s + 48·116-s − 70·121-s − 32·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 1.20·11-s + 3/4·16-s + 1.37·19-s − 1/5·25-s + 2.97·29-s − 1.56·41-s + 1.80·44-s + 2·49-s − 1.04·59-s − 3.58·61-s − 1/4·64-s − 5.22·71-s + 2.06·76-s − 4.05·79-s − 0.635·89-s − 0.299·100-s − 0.398·101-s + 0.766·109-s + 4.45·116-s − 6.36·121-s − 2.86·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 5^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 5^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06341944464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06341944464\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{3} \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T^{2} + 32 T^{3} + 6 T^{4} + 16 T^{5} + 501 T^{6} + 16 p T^{7} + 6 p^{2} T^{8} + 32 p^{3} T^{9} + p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 - 3 T + 24 T^{2} - 25 T^{3} + 24 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| good | 7 | \( ( 1 - p T^{2} + 50 T^{4} - 531 T^{6} + 50 p^{2} T^{8} - p^{5} T^{10} + p^{6} T^{12} )^{2} \) |
| 11 | \( ( 1 - T + 20 T^{2} - 17 T^{3} + 20 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 13 | \( 1 + 66 T^{2} + 2413 T^{4} + 63062 T^{6} + 99698 p T^{8} + 21859066 T^{10} + 309202565 T^{12} + 21859066 p^{2} T^{14} + 99698 p^{5} T^{16} + 63062 p^{6} T^{18} + 2413 p^{8} T^{20} + 66 p^{10} T^{22} + p^{12} T^{24} \) |
| 17 | \( 1 + 58 T^{2} + 1421 T^{4} + 32966 T^{6} + 901610 T^{8} + 17162838 T^{10} + 265402029 T^{12} + 17162838 p^{2} T^{14} + 901610 p^{4} T^{16} + 32966 p^{6} T^{18} + 1421 p^{8} T^{20} + 58 p^{10} T^{22} + p^{12} T^{24} \) |
| 23 | \( 1 - 57 T^{2} + 1051 T^{4} - 14620 T^{6} + 514481 T^{8} - 5020283 T^{10} - 107097658 T^{12} - 5020283 p^{2} T^{14} + 514481 p^{4} T^{16} - 14620 p^{6} T^{18} + 1051 p^{8} T^{20} - 57 p^{10} T^{22} + p^{12} T^{24} \) |
| 29 | \( ( 1 - 8 T - 7 T^{2} + 100 T^{3} + 310 T^{4} + 2502 T^{5} - 39679 T^{6} + 2502 p T^{7} + 310 p^{2} T^{8} + 100 p^{3} T^{9} - 7 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 31 | \( ( 1 + 77 T^{2} + 16 T^{3} + 77 p T^{4} + p^{3} T^{6} )^{4} \) |
| 37 | \( ( 1 - 195 T^{2} + 16646 T^{4} - 799199 T^{6} + 16646 p^{2} T^{8} - 195 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 41 | \( ( 1 + 5 T - 27 T^{2} - 838 T^{3} - 53 p T^{4} + 15633 T^{5} + 286354 T^{6} + 15633 p T^{7} - 53 p^{3} T^{8} - 838 p^{3} T^{9} - 27 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 43 | \( 1 + 178 T^{2} + 16301 T^{4} + 1091270 T^{6} + 61619066 T^{8} + 3149941242 T^{10} + 144853393557 T^{12} + 3149941242 p^{2} T^{14} + 61619066 p^{4} T^{16} + 1091270 p^{6} T^{18} + 16301 p^{8} T^{20} + 178 p^{10} T^{22} + p^{12} T^{24} \) |
| 47 | \( 1 + 18 T^{2} - 3035 T^{4} - 58474 T^{6} + 2792426 T^{8} + 20466154 T^{10} - 1410027091 T^{12} + 20466154 p^{2} T^{14} + 2792426 p^{4} T^{16} - 58474 p^{6} T^{18} - 3035 p^{8} T^{20} + 18 p^{10} T^{22} + p^{12} T^{24} \) |
| 53 | \( 1 + 111 T^{2} + 6051 T^{4} + 54868 T^{6} - 13880691 T^{8} - 1024261299 T^{10} - 60566886642 T^{12} - 1024261299 p^{2} T^{14} - 13880691 p^{4} T^{16} + 54868 p^{6} T^{18} + 6051 p^{8} T^{20} + 111 p^{10} T^{22} + p^{12} T^{24} \) |
| 59 | \( ( 1 + 4 T - 73 T^{2} + 316 T^{3} + 2590 T^{4} - 26304 T^{5} - 111877 T^{6} - 26304 p T^{7} + 2590 p^{2} T^{8} + 316 p^{3} T^{9} - 73 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 61 | \( ( 1 + 14 T + 31 T^{2} - 550 T^{3} - 2860 T^{4} + 6224 T^{5} + 108821 T^{6} + 6224 p T^{7} - 2860 p^{2} T^{8} - 550 p^{3} T^{9} + 31 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 67 | \( 1 + 194 T^{2} + 11805 T^{4} + 1009630 T^{6} + 163131554 T^{8} + 10601096334 T^{10} + 430434923317 T^{12} + 10601096334 p^{2} T^{14} + 163131554 p^{4} T^{16} + 1009630 p^{6} T^{18} + 11805 p^{8} T^{20} + 194 p^{10} T^{22} + p^{12} T^{24} \) |
| 71 | \( ( 1 + 22 T + 133 T^{2} + 1062 T^{3} + 25020 T^{4} + 183208 T^{5} + 617607 T^{6} + 183208 p T^{7} + 25020 p^{2} T^{8} + 1062 p^{3} T^{9} + 133 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 73 | \( 1 + 258 T^{2} + 33409 T^{4} + 2866118 T^{6} + 188835542 T^{8} + 10449018838 T^{10} + 642415211525 T^{12} + 10449018838 p^{2} T^{14} + 188835542 p^{4} T^{16} + 2866118 p^{6} T^{18} + 33409 p^{8} T^{20} + 258 p^{10} T^{22} + p^{12} T^{24} \) |
| 79 | \( ( 1 + 18 T - 5 T^{2} + 26 T^{3} + 29186 T^{4} + 146554 T^{5} - 896341 T^{6} + 146554 p T^{7} + 29186 p^{2} T^{8} + 26 p^{3} T^{9} - 5 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 83 | \( ( 1 - 370 T^{2} + 63911 T^{4} - 6638556 T^{6} + 63911 p^{2} T^{8} - 370 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 89 | \( ( 1 + 3 T + 19 T^{2} + 2968 T^{3} + 5487 T^{4} + 44093 T^{5} + 3576750 T^{6} + 44093 p T^{7} + 5487 p^{2} T^{8} + 2968 p^{3} T^{9} + 19 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 97 | \( 1 + 382 T^{2} + 74889 T^{4} + 10425050 T^{6} + 1185931790 T^{8} + 116693583282 T^{10} + 11078552578789 T^{12} + 116693583282 p^{2} T^{14} + 1185931790 p^{4} T^{16} + 10425050 p^{6} T^{18} + 74889 p^{8} T^{20} + 382 p^{10} T^{22} + p^{12} T^{24} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.81324716462591959709735815705, −2.78157367021325923165354555510, −2.67022619233284728478833955360, −2.64823842282531094341512904435, −2.62820841457768282751685462323, −2.58175110487369703960140711710, −2.38168008145033846846589402729, −2.29784081609639247279238622555, −2.26659533989507826961584332687, −2.20707703192014681131640340920, −1.83824564369291065500963273986, −1.68402018433521256971615163100, −1.60242303478553196694645798307, −1.54488588879048988891144593079, −1.43553961211270615507582537138, −1.37960991791365526001844095743, −1.37582706458850609606939273583, −1.36904253788600550625296989211, −1.32139572660904215133971814284, −1.00812927847500369986293703028, −0.960762116046706131917259293940, −0.51671447148421318941102089980, −0.50817493541554836523279364139, −0.27726018949722587497781034509, −0.01457069423032331164355709470,
0.01457069423032331164355709470, 0.27726018949722587497781034509, 0.50817493541554836523279364139, 0.51671447148421318941102089980, 0.960762116046706131917259293940, 1.00812927847500369986293703028, 1.32139572660904215133971814284, 1.36904253788600550625296989211, 1.37582706458850609606939273583, 1.37960991791365526001844095743, 1.43553961211270615507582537138, 1.54488588879048988891144593079, 1.60242303478553196694645798307, 1.68402018433521256971615163100, 1.83824564369291065500963273986, 2.20707703192014681131640340920, 2.26659533989507826961584332687, 2.29784081609639247279238622555, 2.38168008145033846846589402729, 2.58175110487369703960140711710, 2.62820841457768282751685462323, 2.64823842282531094341512904435, 2.67022619233284728478833955360, 2.78157367021325923165354555510, 2.81324716462591959709735815705
Plot not available for L-functions of degree greater than 10.
