| L(s) = 1 | + 5·3-s + 3·5-s + 9·7-s + 3·9-s + 80·11-s + 26·13-s + 15·15-s + 19·17-s − 84·19-s + 45·21-s − 196·23-s − 239·25-s + 40·27-s + 44·29-s + 86·31-s + 400·33-s + 27·35-s − 209·37-s + 130·39-s − 230·41-s + 287·43-s + 9·45-s − 435·47-s − 111·49-s + 95·51-s + 118·53-s + 240·55-s + ⋯ |
| L(s) = 1 | + 0.962·3-s + 0.268·5-s + 0.485·7-s + 1/9·9-s + 2.19·11-s + 0.554·13-s + 0.258·15-s + 0.271·17-s − 1.01·19-s + 0.467·21-s − 1.77·23-s − 1.91·25-s + 0.285·27-s + 0.281·29-s + 0.498·31-s + 2.11·33-s + 0.130·35-s − 0.928·37-s + 0.533·39-s − 0.876·41-s + 1.01·43-s + 0.0298·45-s − 1.35·47-s − 0.323·49-s + 0.260·51-s + 0.305·53-s + 0.588·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 692224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 692224 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.042969038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.042969038\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 5 T + 22 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 3 T + 248 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 9 T + 192 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 80 T + 3650 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 19 T + 8688 T^{2} - 19 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 84 T + 11130 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 196 T + 33326 T^{2} + 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 44 T + 10094 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 86 T + 56518 T^{2} - 86 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 209 T + 112120 T^{2} + 209 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 230 T + 149010 T^{2} + 230 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 287 T + 92698 T^{2} - 287 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 435 T + 192728 T^{2} + 435 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 118 T + 297410 T^{2} - 118 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 368 T + 379266 T^{2} + 368 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1058 T + 580378 T^{2} - 1058 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 68 T + 373930 T^{2} - 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 131 T + 493328 T^{2} - 131 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 456 T + 542718 T^{2} - 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1008 T + 1233294 T^{2} - 1008 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1958 T + 1961238 T^{2} - 1958 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 720 T + 899726 T^{2} + 720 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 928 T + 943870 T^{2} + 928 p^{3} T^{3} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.763049662525446395019995029923, −9.712590921001709041731373922944, −9.087716683540676492464833672287, −8.808469024599310440315613518188, −8.259484174151348598214097865757, −8.104213480782948364330740087994, −7.74037628996073024229315530250, −6.87647617780041753909116587389, −6.51438188037783343215244016209, −6.32278332797021186045669683889, −5.71031202655987873146794101578, −5.20997638165921646458055221059, −4.26840125104223948878269564929, −4.20714852400493968454377521911, −3.57017249181654332997882054386, −3.29542244590481286008132887937, −2.08723788375339219636982719621, −2.07823039899284896462840344457, −1.40286061701553153146687171795, −0.55174718232460916016725526843,
0.55174718232460916016725526843, 1.40286061701553153146687171795, 2.07823039899284896462840344457, 2.08723788375339219636982719621, 3.29542244590481286008132887937, 3.57017249181654332997882054386, 4.20714852400493968454377521911, 4.26840125104223948878269564929, 5.20997638165921646458055221059, 5.71031202655987873146794101578, 6.32278332797021186045669683889, 6.51438188037783343215244016209, 6.87647617780041753909116587389, 7.74037628996073024229315530250, 8.104213480782948364330740087994, 8.259484174151348598214097865757, 8.808469024599310440315613518188, 9.087716683540676492464833672287, 9.712590921001709041731373922944, 9.763049662525446395019995029923
