| L(s) = 1 | − 70·5-s + 1.11e3·7-s + 7.25e3·11-s + 1.37e4·13-s − 1.69e4·17-s + 3.40e4·19-s + 3.23e4·23-s − 7.32e4·25-s − 3.41e4·29-s − 1.20e5·31-s − 7.80e4·35-s + 3.52e4·37-s + 4.84e5·41-s − 6.72e5·43-s + 1.20e6·47-s + 4.20e5·49-s − 8.51e5·53-s − 5.07e5·55-s − 6.95e5·59-s + 7.16e4·61-s − 9.63e5·65-s − 3.07e5·67-s + 7.57e5·71-s + 3.91e6·73-s + 8.08e6·77-s + 3.14e5·79-s + 1.53e6·83-s + ⋯ |
| L(s) = 1 | − 0.250·5-s + 1.22·7-s + 1.64·11-s + 1.73·13-s − 0.838·17-s + 1.13·19-s + 0.554·23-s − 0.937·25-s − 0.260·29-s − 0.726·31-s − 0.307·35-s + 0.114·37-s + 1.09·41-s − 1.28·43-s + 1.69·47-s + 0.510·49-s − 0.785·53-s − 0.411·55-s − 0.440·59-s + 0.0404·61-s − 0.434·65-s − 0.124·67-s + 0.251·71-s + 1.17·73-s + 2.01·77-s + 0.0717·79-s + 0.294·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.172387106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.172387106\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 70T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.11e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 7.25e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.37e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.69e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.40e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.23e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 3.41e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.20e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.52e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.84e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.72e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.20e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 8.51e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 6.95e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 7.16e4T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.07e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 7.57e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.91e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.14e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.53e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.51e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 5.00e4T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.04161748164557545228735591588, −9.405072714659069593291113179762, −8.723931549655372883833385107403, −7.77838689744546476292468694066, −6.67229038458026723345184865805, −5.63698622448832598930931237535, −4.34379630453734626422693160392, −3.55887656514292189950855260125, −1.75459166524746513789779276498, −0.982972969137827406504974912180,
0.982972969137827406504974912180, 1.75459166524746513789779276498, 3.55887656514292189950855260125, 4.34379630453734626422693160392, 5.63698622448832598930931237535, 6.67229038458026723345184865805, 7.77838689744546476292468694066, 8.723931549655372883833385107403, 9.405072714659069593291113179762, 11.04161748164557545228735591588
