| L(s) = 1 | − 30.9·2-s + 166.·3-s + 443.·4-s + 268.·5-s − 5.14e3·6-s − 2.40e3·7-s + 2.11e3·8-s + 7.97e3·9-s − 8.28e3·10-s + 4.12e4·11-s + 7.37e4·12-s − 2.85e4·13-s + 7.42e4·14-s + 4.45e4·15-s − 2.92e5·16-s − 2.51e5·17-s − 2.46e5·18-s − 6.83e5·19-s + 1.18e5·20-s − 3.99e5·21-s − 1.27e6·22-s + 1.56e6·23-s + 3.51e5·24-s − 1.88e6·25-s + 8.82e5·26-s − 1.94e6·27-s − 1.06e6·28-s + ⋯ |
| L(s) = 1 | − 1.36·2-s + 1.18·3-s + 0.866·4-s + 0.191·5-s − 1.61·6-s − 0.377·7-s + 0.182·8-s + 0.404·9-s − 0.262·10-s + 0.849·11-s + 1.02·12-s − 0.277·13-s + 0.516·14-s + 0.227·15-s − 1.11·16-s − 0.729·17-s − 0.553·18-s − 1.20·19-s + 0.166·20-s − 0.448·21-s − 1.16·22-s + 1.16·23-s + 0.215·24-s − 0.963·25-s + 0.378·26-s − 0.705·27-s − 0.327·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 + 2.40e3T \) |
| 13 | \( 1 + 2.85e4T \) |
| good | 2 | \( 1 + 30.9T + 512T^{2} \) |
| 3 | \( 1 - 166.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 268.T + 1.95e6T^{2} \) |
| 11 | \( 1 - 4.12e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 2.51e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 6.83e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.56e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.74e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.17e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.36e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.25e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.35e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.04e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 8.70e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 9.52e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.57e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.45e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.46e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.96e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.42e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.32e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.24e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.68e8T + 7.60e17T^{2} \) |
| 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
−11.35664355001734283052386866260, −10.18452349507920007514711141853, −9.166740755740146764819033779363, −8.738342548917448527147923363784, −7.60842072332714722031381566793, −6.46846161953902000985113977909, −4.26131706252149019653068046002, −2.69657273036469731976078711485, −1.56417829322287768015129647834, 0,
1.56417829322287768015129647834, 2.69657273036469731976078711485, 4.26131706252149019653068046002, 6.46846161953902000985113977909, 7.60842072332714722031381566793, 8.738342548917448527147923363784, 9.166740755740146764819033779363, 10.18452349507920007514711141853, 11.35664355001734283052386866260
