| L(s) = 1 | + 4-s + 6·11-s − 4·17-s + 12·19-s + 12·25-s − 6·29-s − 6·37-s − 2·43-s + 6·44-s + 49-s + 28·53-s − 24·59-s + 20·61-s − 64-s − 24·67-s − 4·68-s − 18·71-s + 12·76-s − 12·79-s − 12·89-s + 6·97-s + 12·100-s + 20·101-s + 28·103-s − 10·107-s − 10·113-s − 6·116-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.80·11-s − 0.970·17-s + 2.75·19-s + 12/5·25-s − 1.11·29-s − 0.986·37-s − 0.304·43-s + 0.904·44-s + 1/7·49-s + 3.84·53-s − 3.12·59-s + 2.56·61-s − 1/8·64-s − 2.93·67-s − 0.485·68-s − 2.13·71-s + 1.37·76-s − 1.35·79-s − 1.27·89-s + 0.609·97-s + 6/5·100-s + 1.99·101-s + 2.75·103-s − 0.966·107-s − 0.940·113-s − 0.557·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.435159406\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.435159406\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 74 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 6 T + 3 p T^{2} - 126 T^{3} + 452 T^{4} - 126 p T^{5} + 3 p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 4 T - 19 T^{2} + 4 T^{3} + 664 T^{4} + 4 p T^{5} - 19 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 12 T + 97 T^{2} - 588 T^{3} + 2952 T^{4} - 588 p T^{5} + 97 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 - 34 T^{2} + 627 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 28 T^{2} + 1818 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 6 T + 80 T^{2} + 408 T^{3} + 3699 T^{4} + 408 p T^{5} + 80 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 33 T^{2} - 592 T^{4} + 33 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 2 T - 80 T^{2} - 4 T^{3} + 5035 T^{4} - 4 p T^{5} - 80 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 162 T^{2} + 10931 T^{4} - 162 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 24 T + 322 T^{2} + 3120 T^{3} + 24747 T^{4} + 3120 p T^{5} + 322 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 20 T + 181 T^{2} - 1940 T^{3} + 19840 T^{4} - 1940 p T^{5} + 181 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 24 T + 338 T^{2} + 3504 T^{3} + 29691 T^{4} + 3504 p T^{5} + 338 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 18 T + 268 T^{2} + 2880 T^{3} + 28227 T^{4} + 2880 p T^{5} + 268 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 7014 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 17802 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T + 229 T^{2} + 2172 T^{3} + 29112 T^{4} + 2172 p T^{5} + 229 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 6 T + 208 T^{2} - 1176 T^{3} + 30171 T^{4} - 1176 p T^{5} + 208 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.75576668788393758621570743412, −6.66034027902682380566389683399, −6.11184067962316252122264481034, −6.04240685948701470741419830826, −5.98056273394460709791438895487, −5.70332936247945178580541172926, −5.28515527781232310540471930832, −5.15154076030173980023054868720, −5.00341462446679008297272257898, −4.78198498924502143238538609453, −4.48742666992015819705964100624, −4.23679128141635827233325340439, −3.95922014038987569992646681516, −3.63783633997904440292916259931, −3.53604627461980596669166654084, −3.43688039242239385669896405475, −2.91509341430353355941706197240, −2.69507613829061416143134027523, −2.51548100720292289646161254086, −2.35630445313236814519010726077, −1.58045620115064456489568869081, −1.50015895838871874973977241559, −1.21094993674578404159716550300, −1.09635316575717931541341204383, −0.32238785450571022440084408146,
0.32238785450571022440084408146, 1.09635316575717931541341204383, 1.21094993674578404159716550300, 1.50015895838871874973977241559, 1.58045620115064456489568869081, 2.35630445313236814519010726077, 2.51548100720292289646161254086, 2.69507613829061416143134027523, 2.91509341430353355941706197240, 3.43688039242239385669896405475, 3.53604627461980596669166654084, 3.63783633997904440292916259931, 3.95922014038987569992646681516, 4.23679128141635827233325340439, 4.48742666992015819705964100624, 4.78198498924502143238538609453, 5.00341462446679008297272257898, 5.15154076030173980023054868720, 5.28515527781232310540471930832, 5.70332936247945178580541172926, 5.98056273394460709791438895487, 6.04240685948701470741419830826, 6.11184067962316252122264481034, 6.66034027902682380566389683399, 6.75576668788393758621570743412
