| L(s) = 1 | − 3·3-s − 8·4-s + 12·5-s − 2·7-s + 9·9-s + 36·11-s + 24·12-s − 36·15-s + 64·16-s − 78·17-s − 74·19-s − 96·20-s + 6·21-s − 96·23-s + 19·25-s − 27·27-s + 16·28-s + 18·29-s + 214·31-s − 108·33-s − 24·35-s − 72·36-s + 286·37-s + 384·41-s + 524·43-s − 288·44-s + 108·45-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 1.07·5-s − 0.107·7-s + 1/3·9-s + 0.986·11-s + 0.577·12-s − 0.619·15-s + 16-s − 1.11·17-s − 0.893·19-s − 1.07·20-s + 0.0623·21-s − 0.870·23-s + 0.151·25-s − 0.192·27-s + 0.107·28-s + 0.115·29-s + 1.23·31-s − 0.569·33-s − 0.115·35-s − 1/3·36-s + 1.27·37-s + 1.46·41-s + 1.85·43-s − 0.986·44-s + 0.357·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.425778413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.425778413\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p^{3} T^{2} \) |
| 5 | \( 1 - 12 T + p^{3} T^{2} \) |
| 7 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 17 | \( 1 + 78 T + p^{3} T^{2} \) |
| 19 | \( 1 + 74 T + p^{3} T^{2} \) |
| 23 | \( 1 + 96 T + p^{3} T^{2} \) |
| 29 | \( 1 - 18 T + p^{3} T^{2} \) |
| 31 | \( 1 - 214 T + p^{3} T^{2} \) |
| 37 | \( 1 - 286 T + p^{3} T^{2} \) |
| 41 | \( 1 - 384 T + p^{3} T^{2} \) |
| 43 | \( 1 - 524 T + p^{3} T^{2} \) |
| 47 | \( 1 + 300 T + p^{3} T^{2} \) |
| 53 | \( 1 - 558 T + p^{3} T^{2} \) |
| 59 | \( 1 + 576 T + p^{3} T^{2} \) |
| 61 | \( 1 - 74 T + p^{3} T^{2} \) |
| 67 | \( 1 + 38 T + p^{3} T^{2} \) |
| 71 | \( 1 - 456 T + p^{3} T^{2} \) |
| 73 | \( 1 - 682 T + p^{3} T^{2} \) |
| 79 | \( 1 - 704 T + p^{3} T^{2} \) |
| 83 | \( 1 - 888 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1020 T + p^{3} T^{2} \) |
| 97 | \( 1 + 110 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.35799618265716317387053278380, −9.540365643710024220110592462692, −9.024165108235927590371941439577, −7.903894330848768917816924189738, −6.37086881826297221728138093553, −6.04387648420180379982702614490, −4.73841640533161114360269252364, −4.01776500521957705696944555852, −2.20404805590089038910357439652, −0.77878815646777223540205976655,
0.77878815646777223540205976655, 2.20404805590089038910357439652, 4.01776500521957705696944555852, 4.73841640533161114360269252364, 6.04387648420180379982702614490, 6.37086881826297221728138093553, 7.903894330848768917816924189738, 9.024165108235927590371941439577, 9.540365643710024220110592462692, 10.35799618265716317387053278380
