| L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s + 10.4·5-s + 36·6-s + 172.·7-s + 64·8-s + 81·9-s + 41.6·10-s + 483.·11-s + 144·12-s + 1.07e3·13-s + 688.·14-s + 93.7·15-s + 256·16-s − 62.0·17-s + 324·18-s − 2.88e3·19-s + 166.·20-s + 1.54e3·21-s + 1.93e3·22-s − 2.80e3·23-s + 576·24-s − 3.01e3·25-s + 4.31e3·26-s + 729·27-s + 2.75e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.186·5-s + 0.408·6-s + 1.32·7-s + 0.353·8-s + 0.333·9-s + 0.131·10-s + 1.20·11-s + 0.288·12-s + 1.76·13-s + 0.938·14-s + 0.107·15-s + 0.250·16-s − 0.0521·17-s + 0.235·18-s − 1.83·19-s + 0.0931·20-s + 0.766·21-s + 0.851·22-s − 1.10·23-s + 0.204·24-s − 0.965·25-s + 1.25·26-s + 0.192·27-s + 0.663·28-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
| \(L(3)\) |
\(\approx\) |
\(5.700870640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.700870640\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 10.4T + 3.12e3T^{2} \) |
| 7 | \( 1 - 172.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 483.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.07e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 62.0T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.88e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.80e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.10e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.06e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.34e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.63e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.50e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.56e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.38e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 4.95e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 8.77e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.92e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.17e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.04e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.71e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.13e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.18e5T + 8.58e9T^{2} \) |
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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
−11.03926680613742391517111746022, −9.704742497899897466388488197032, −8.496433626647986913969793172209, −8.051046225400137873738327046894, −6.57783582887002405649238949528, −5.85165835161394143526475321648, −4.30494562370634849902074123952, −3.87352826566234912051338749244, −2.14963118157893043973125138763, −1.34255568892018872948357901730,
1.34255568892018872948357901730, 2.14963118157893043973125138763, 3.87352826566234912051338749244, 4.30494562370634849902074123952, 5.85165835161394143526475321648, 6.57783582887002405649238949528, 8.051046225400137873738327046894, 8.496433626647986913969793172209, 9.704742497899897466388488197032, 11.03926680613742391517111746022
