| L(s) = 1 | − 4.79·2-s + 15.0·4-s − 7.96·5-s − 5.93·7-s − 33.7·8-s + 38.2·10-s − 14.7·11-s + 19.5·13-s + 28.4·14-s + 41.7·16-s − 50.1·17-s + 139.·19-s − 119.·20-s + 71.0·22-s + 130.·23-s − 61.4·25-s − 94.0·26-s − 89.1·28-s − 204.·29-s + 141.·31-s + 69.7·32-s + 240.·34-s + 47.2·35-s + 300.·37-s − 671.·38-s + 269.·40-s + 7.52·41-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 1.87·4-s − 0.712·5-s − 0.320·7-s − 1.49·8-s + 1.20·10-s − 0.405·11-s + 0.418·13-s + 0.543·14-s + 0.652·16-s − 0.715·17-s + 1.68·19-s − 1.33·20-s + 0.688·22-s + 1.18·23-s − 0.491·25-s − 0.709·26-s − 0.601·28-s − 1.30·29-s + 0.819·31-s + 0.385·32-s + 1.21·34-s + 0.228·35-s + 1.33·37-s − 2.86·38-s + 1.06·40-s + 0.0286·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5552308434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5552308434\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 + 4.79T + 8T^{2} \) |
| 5 | \( 1 + 7.96T + 125T^{2} \) |
| 7 | \( 1 + 5.93T + 343T^{2} \) |
| 11 | \( 1 + 14.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 19.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 50.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 139.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 130.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 204.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 141.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 300.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 7.52T + 6.89e4T^{2} \) |
| 43 | \( 1 - 26.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + 495.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 430.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 200.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 269.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 162.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 639.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 312.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 396.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 451.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 561.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.45e3T + 9.12e5T^{2} \) |
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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
−8.751777617115260586342468223464, −7.984629518177417055431117380365, −7.46952201844699269819744817322, −6.80300549687712470449316065044, −5.85505865460796941571116199948, −4.71783164406601817983616529365, −3.49800304360993889378525227040, −2.62520554031343178948841322226, −1.39838864705343124251181914211, −0.46867812638823540290745033186,
0.46867812638823540290745033186, 1.39838864705343124251181914211, 2.62520554031343178948841322226, 3.49800304360993889378525227040, 4.71783164406601817983616529365, 5.85505865460796941571116199948, 6.80300549687712470449316065044, 7.46952201844699269819744817322, 7.984629518177417055431117380365, 8.751777617115260586342468223464
