| L(s) = 1 | + 7.32·2-s − 458.·4-s + 625·5-s − 2.40e3·7-s − 7.10e3·8-s + 4.57e3·10-s + 2.55e4·11-s − 3.67e4·13-s − 1.75e4·14-s + 1.82e5·16-s − 1.91e5·17-s + 5.38e5·19-s − 2.86e5·20-s + 1.87e5·22-s − 1.45e6·23-s + 3.90e5·25-s − 2.68e5·26-s + 1.10e6·28-s + 9.95e5·29-s + 3.11e6·31-s + 4.97e6·32-s − 1.40e6·34-s − 1.50e6·35-s + 6.55e6·37-s + 3.94e6·38-s − 4.44e6·40-s + 2.64e6·41-s + ⋯ |
| L(s) = 1 | + 0.323·2-s − 0.895·4-s + 0.447·5-s − 0.377·7-s − 0.613·8-s + 0.144·10-s + 0.526·11-s − 0.356·13-s − 0.122·14-s + 0.696·16-s − 0.555·17-s + 0.948·19-s − 0.400·20-s + 0.170·22-s − 1.08·23-s + 0.200·25-s − 0.115·26-s + 0.338·28-s + 0.261·29-s + 0.605·31-s + 0.839·32-s − 0.179·34-s − 0.169·35-s + 0.575·37-s + 0.307·38-s − 0.274·40-s + 0.146·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 625T \) |
| 7 | \( 1 + 2.40e3T \) |
| good | 2 | \( 1 - 7.32T + 512T^{2} \) |
| 11 | \( 1 - 2.55e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 3.67e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.91e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.38e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.45e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 9.95e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.11e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 6.55e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.64e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 7.42e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.64e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.36e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 5.20e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 6.55e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.13e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.26e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 9.95e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.03e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.47e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.33e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.53e8T + 7.60e17T^{2} \) |
| 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
−9.588752522782311944610143092436, −8.892662454489923384050515387531, −7.80188897958582893949928518553, −6.53272801875708598740682084733, −5.66810737170310285334505618964, −4.65058392081981738471723867619, −3.71970605952549733346946210914, −2.57363592073880507451737530294, −1.13954627883648156359915491816, 0,
1.13954627883648156359915491816, 2.57363592073880507451737530294, 3.71970605952549733346946210914, 4.65058392081981738471723867619, 5.66810737170310285334505618964, 6.53272801875708598740682084733, 7.80188897958582893949928518553, 8.892662454489923384050515387531, 9.588752522782311944610143092436
