| L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s + 107.·5-s − 36·6-s − 223.·7-s + 64·8-s + 81·9-s + 430.·10-s + 283.·11-s − 144·12-s − 1.08e3·13-s − 893.·14-s − 968.·15-s + 256·16-s + 1.93e3·17-s + 324·18-s + 2.46e3·19-s + 1.72e3·20-s + 2.01e3·21-s + 1.13e3·22-s − 1.89e3·23-s − 576·24-s + 8.46e3·25-s − 4.34e3·26-s − 729·27-s − 3.57e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.92·5-s − 0.408·6-s − 1.72·7-s + 0.353·8-s + 0.333·9-s + 1.36·10-s + 0.706·11-s − 0.288·12-s − 1.78·13-s − 1.21·14-s − 1.11·15-s + 0.250·16-s + 1.62·17-s + 0.235·18-s + 1.56·19-s + 0.962·20-s + 0.995·21-s + 0.499·22-s − 0.748·23-s − 0.204·24-s + 2.70·25-s − 1.26·26-s − 0.192·27-s − 0.861·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.492180751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.492180751\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 + 9T \) |
| 59 | \( 1 + 3.48e3T \) |
| good | 5 | \( 1 - 107.T + 3.12e3T^{2} \) |
| 7 | \( 1 + 223.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 283.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.08e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.93e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.46e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.89e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.91e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 768.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.35e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 8.18e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.56e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 369.T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.24e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 3.55e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 7.74e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.49e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.38e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.08e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.63e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.74e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.08e4T + 8.58e9T^{2} \) |
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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
−10.19131953995642871216007822176, −9.952663617254530661813301574266, −9.356622161100645015476497465617, −7.26270146521607143377666618669, −6.51751471411509824936779401253, −5.72280011279623253978913773985, −5.10960563210799114091992517472, −3.37111749574824606022592761648, −2.40554528662279928294378248363, −0.966697643504649610520063055262,
0.966697643504649610520063055262, 2.40554528662279928294378248363, 3.37111749574824606022592761648, 5.10960563210799114091992517472, 5.72280011279623253978913773985, 6.51751471411509824936779401253, 7.26270146521607143377666618669, 9.356622161100645015476497465617, 9.952663617254530661813301574266, 10.19131953995642871216007822176
