| L(s) = 1 | + 524.·2-s + 8.85e3·3-s + 1.44e5·4-s + 3.90e5·5-s + 4.64e6·6-s + 1.13e7·7-s + 6.96e6·8-s − 5.08e7·9-s + 2.04e8·10-s + 3.42e7·11-s + 1.27e9·12-s + 4.29e9·13-s + 5.93e9·14-s + 3.45e9·15-s − 1.52e10·16-s − 4.75e10·17-s − 2.66e10·18-s − 7.29e9·19-s + 5.63e10·20-s + 1.00e11·21-s + 1.79e10·22-s − 5.76e11·23-s + 6.16e10·24-s + 1.52e11·25-s + 2.25e12·26-s − 1.59e12·27-s + 1.63e12·28-s + ⋯ |
| L(s) = 1 | + 1.44·2-s + 0.778·3-s + 1.10·4-s + 0.447·5-s + 1.12·6-s + 0.741·7-s + 0.146·8-s − 0.393·9-s + 0.648·10-s + 0.0481·11-s + 0.857·12-s + 1.46·13-s + 1.07·14-s + 0.348·15-s − 0.888·16-s − 1.65·17-s − 0.570·18-s − 0.0984·19-s + 0.492·20-s + 0.577·21-s + 0.0698·22-s − 1.53·23-s + 0.114·24-s + 0.200·25-s + 2.11·26-s − 1.08·27-s + 0.817·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(4.557206687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.557206687\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 3.90e5T \) |
| good | 2 | \( 1 - 524.T + 1.31e5T^{2} \) |
| 3 | \( 1 - 8.85e3T + 1.29e8T^{2} \) |
| 7 | \( 1 - 1.13e7T + 2.32e14T^{2} \) |
| 11 | \( 1 - 3.42e7T + 5.05e17T^{2} \) |
| 13 | \( 1 - 4.29e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 4.75e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 7.29e9T + 5.48e21T^{2} \) |
| 23 | \( 1 + 5.76e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 1.27e12T + 7.25e24T^{2} \) |
| 31 | \( 1 - 6.09e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 9.62e12T + 4.56e26T^{2} \) |
| 41 | \( 1 - 6.45e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 5.47e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 2.90e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 4.38e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 3.50e14T + 1.27e30T^{2} \) |
| 61 | \( 1 - 2.64e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 2.24e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 1.01e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 2.73e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 9.55e14T + 1.81e32T^{2} \) |
| 83 | \( 1 + 2.07e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 3.36e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 4.29e16T + 5.95e33T^{2} \) |
| 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
−20.19053054598510635705706715994, −17.93774134348699423476265093699, −15.53797070347130403206545765411, −14.18050414684220991636889894747, −13.36796168767200125607593882349, −11.36426146000042134445895139680, −8.639285122035680608624750600428, −6.01966647911643016894700672962, −4.09724958081888545131008704702, −2.29980625798811169293843238338,
2.29980625798811169293843238338, 4.09724958081888545131008704702, 6.01966647911643016894700672962, 8.639285122035680608624750600428, 11.36426146000042134445895139680, 13.36796168767200125607593882349, 14.18050414684220991636889894747, 15.53797070347130403206545765411, 17.93774134348699423476265093699, 20.19053054598510635705706715994
