| L(s) = 1 | + 15.4·2-s + 111.·4-s − 125·5-s − 275.·7-s − 249.·8-s − 1.93e3·10-s − 6.77e3·11-s − 305.·13-s − 4.26e3·14-s − 1.81e4·16-s + 3.95e3·17-s + 1.14e4·19-s − 1.39e4·20-s − 1.04e5·22-s + 5.91e4·23-s + 1.56e4·25-s − 4.72e3·26-s − 3.08e4·28-s + 4.71e4·29-s + 1.42e5·31-s − 2.49e5·32-s + 6.13e4·34-s + 3.44e4·35-s + 1.95e5·37-s + 1.77e5·38-s + 3.11e4·40-s + 7.55e5·41-s + ⋯ |
| L(s) = 1 | + 1.36·2-s + 0.874·4-s − 0.447·5-s − 0.303·7-s − 0.172·8-s − 0.612·10-s − 1.53·11-s − 0.0385·13-s − 0.415·14-s − 1.10·16-s + 0.195·17-s + 0.383·19-s − 0.390·20-s − 2.09·22-s + 1.01·23-s + 0.199·25-s − 0.0527·26-s − 0.265·28-s + 0.359·29-s + 0.859·31-s − 1.34·32-s + 0.267·34-s + 0.135·35-s + 0.634·37-s + 0.524·38-s + 0.0770·40-s + 1.71·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.112217910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.112217910\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 125T \) |
| good | 2 | \( 1 - 15.4T + 128T^{2} \) |
| 7 | \( 1 + 275.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 6.77e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 305.T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.95e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.14e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.91e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 4.71e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.42e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.95e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.55e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.84e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.34e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.12e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 6.16e4T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.60e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 5.46e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.46e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.00e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.30e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.06e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 7.72e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 7.73e6T + 8.07e13T^{2} \) |
| 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
−10.31794132916718859303325732014, −9.173528668076707441809740373406, −8.028001965335775026849952822703, −7.11209362788075227616584035111, −5.99674101297303502546522377079, −5.14677135354617811117604532852, −4.35769996685733124494215442300, −3.18074084955418915625275850677, −2.53906999691078669705723719651, −0.63325329353154947586872499703,
0.63325329353154947586872499703, 2.53906999691078669705723719651, 3.18074084955418915625275850677, 4.35769996685733124494215442300, 5.14677135354617811117604532852, 5.99674101297303502546522377079, 7.11209362788075227616584035111, 8.028001965335775026849952822703, 9.173528668076707441809740373406, 10.31794132916718859303325732014
