| L(s) = 1 | − 37.3·2-s + 245.·3-s + 884.·4-s + 1.54e3·5-s − 9.18e3·6-s + 9.37e3·7-s − 1.39e4·8-s + 4.07e4·9-s − 5.79e4·10-s + 3.18e4·11-s + 2.17e5·12-s − 3.83e3·13-s − 3.50e5·14-s + 3.80e5·15-s + 6.75e4·16-s − 1.52e6·18-s + 7.04e5·19-s + 1.37e6·20-s + 2.30e6·21-s − 1.19e6·22-s + 1.41e5·23-s − 3.42e6·24-s + 4.47e5·25-s + 1.43e5·26-s + 5.16e6·27-s + 8.29e6·28-s − 5.40e6·29-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + 1.75·3-s + 1.72·4-s + 1.10·5-s − 2.89·6-s + 1.47·7-s − 1.20·8-s + 2.06·9-s − 1.83·10-s + 0.655·11-s + 3.02·12-s − 0.0372·13-s − 2.43·14-s + 1.94·15-s + 0.257·16-s − 3.41·18-s + 1.24·19-s + 1.91·20-s + 2.58·21-s − 1.08·22-s + 0.105·23-s − 2.10·24-s + 0.228·25-s + 0.0615·26-s + 1.87·27-s + 2.55·28-s − 1.41·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.657822366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.657822366\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + 37.3T + 512T^{2} \) |
| 3 | \( 1 - 245.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.54e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 9.37e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.18e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 3.83e3T + 1.06e10T^{2} \) |
| 19 | \( 1 - 7.04e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.41e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.40e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.78e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 2.94e5T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.41e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.87e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.19e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 8.33e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.05e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 9.79e6T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.49e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 7.01e6T + 4.58e16T^{2} \) |
| 73 | \( 1 - 9.11e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 8.25e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.73e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.66e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 5.74e8T + 7.60e17T^{2} \) |
| 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
−9.692280370221318721237019942430, −9.220069165367515424695355989590, −8.571273060829532523696648315564, −7.67606437991770963989819099248, −7.11976858934957512788945748996, −5.41720699318872441006093690214, −3.81138165211653792249936450774, −2.33807371178053413600641209844, −1.79633851840638272364486641654, −1.12099766780410136281682382868,
1.12099766780410136281682382868, 1.79633851840638272364486641654, 2.33807371178053413600641209844, 3.81138165211653792249936450774, 5.41720699318872441006093690214, 7.11976858934957512788945748996, 7.67606437991770963989819099248, 8.571273060829532523696648315564, 9.220069165367515424695355989590, 9.692280370221318721237019942430
