| L(s) = 1 | + 13.9·2-s + 65.3·4-s − 125·5-s + 343·7-s − 871.·8-s − 1.73e3·10-s − 2.95e3·11-s + 2.18e3·13-s + 4.76e3·14-s − 2.04e4·16-s − 1.86e3·17-s + 5.03e3·19-s − 8.16e3·20-s − 4.11e4·22-s + 7.34e4·23-s + 1.56e4·25-s + 3.03e4·26-s + 2.24e4·28-s + 5.27e4·29-s − 533.·31-s − 1.73e5·32-s − 2.58e4·34-s − 4.28e4·35-s + 1.36e5·37-s + 7.00e4·38-s + 1.08e5·40-s + 4.52e5·41-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.510·4-s − 0.447·5-s + 0.377·7-s − 0.601·8-s − 0.549·10-s − 0.670·11-s + 0.275·13-s + 0.464·14-s − 1.24·16-s − 0.0918·17-s + 0.168·19-s − 0.228·20-s − 0.823·22-s + 1.25·23-s + 0.199·25-s + 0.338·26-s + 0.192·28-s + 0.401·29-s − 0.00321·31-s − 0.934·32-s − 0.112·34-s − 0.169·35-s + 0.442·37-s + 0.207·38-s + 0.269·40-s + 1.02·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.412833896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.412833896\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 125T \) |
| 7 | \( 1 - 343T \) |
| good | 2 | \( 1 - 13.9T + 128T^{2} \) |
| 11 | \( 1 + 2.95e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 2.18e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.86e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.03e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 7.34e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 5.27e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 533.T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.36e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.52e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.30e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 5.80e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 8.24e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 3.24e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.11e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.52e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.84e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.16e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.05e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.53e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.17e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 5.36e6T + 8.07e13T^{2} \) |
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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
−10.86703988267344349118773625955, −9.479279724782547553809866677483, −8.473154882939797325608130364453, −7.43777779673955467121214966672, −6.30638753816352349449733177675, −5.25212603257841868557773945690, −4.51798267611931395691558274455, −3.45559460087227238768987137991, −2.46173712743842894917951087796, −0.73746575236068409881119912779,
0.73746575236068409881119912779, 2.46173712743842894917951087796, 3.45559460087227238768987137991, 4.51798267611931395691558274455, 5.25212603257841868557773945690, 6.30638753816352349449733177675, 7.43777779673955467121214966672, 8.473154882939797325608130364453, 9.479279724782547553809866677483, 10.86703988267344349118773625955
