| L(s) = 1 | − 4·2-s + 16·4-s − 24.6·5-s + 103.·7-s − 64·8-s + 98.5·10-s − 457.·11-s − 1.02e3·13-s − 412.·14-s + 256·16-s + 303.·17-s + 2.73e3·19-s − 394.·20-s + 1.83e3·22-s − 529·23-s − 2.51e3·25-s + 4.10e3·26-s + 1.65e3·28-s − 596.·29-s + 7.03e3·31-s − 1.02e3·32-s − 1.21e3·34-s − 2.54e3·35-s − 5.33e3·37-s − 1.09e4·38-s + 1.57e3·40-s + 1.26e3·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.440·5-s + 0.795·7-s − 0.353·8-s + 0.311·10-s − 1.14·11-s − 1.68·13-s − 0.562·14-s + 0.250·16-s + 0.254·17-s + 1.73·19-s − 0.220·20-s + 0.806·22-s − 0.208·23-s − 0.805·25-s + 1.18·26-s + 0.397·28-s − 0.131·29-s + 1.31·31-s − 0.176·32-s − 0.180·34-s − 0.350·35-s − 0.640·37-s − 1.22·38-s + 0.155·40-s + 0.117·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.038365847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.038365847\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 + 529T \) |
| good | 5 | \( 1 + 24.6T + 3.12e3T^{2} \) |
| 7 | \( 1 - 103.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 457.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.02e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 303.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.73e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 596.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.03e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.33e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.26e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.36e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.39e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.22e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.06e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.66e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.36e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.14e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.38e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.33e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.91e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.16e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.11e4T + 8.58e9T^{2} \) |
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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
−10.09773806823976174329026749635, −9.775877017216362515823169102913, −8.339590753238997898076642593816, −7.74563021796323353024611678481, −7.10777079026494475917091443180, −5.50206991835791707143121802743, −4.75463556327975743658448397164, −3.12911848362504003587317056732, −2.03807294423683682487089448856, −0.57902739489284359213514593082,
0.57902739489284359213514593082, 2.03807294423683682487089448856, 3.12911848362504003587317056732, 4.75463556327975743658448397164, 5.50206991835791707143121802743, 7.10777079026494475917091443180, 7.74563021796323353024611678481, 8.339590753238997898076642593816, 9.775877017216362515823169102913, 10.09773806823976174329026749635
