| L(s) = 1 | − 0.429·2-s − 22.9·3-s − 31.8·4-s − 25·5-s + 9.85·6-s − 78.3·7-s + 27.4·8-s + 283.·9-s + 10.7·10-s − 121·11-s + 730.·12-s + 215.·13-s + 33.6·14-s + 573.·15-s + 1.00e3·16-s − 352.·17-s − 121.·18-s + 361·19-s + 795.·20-s + 1.79e3·21-s + 51.9·22-s − 427.·23-s − 629.·24-s + 625·25-s − 92.4·26-s − 935.·27-s + 2.49e3·28-s + ⋯ |
| L(s) = 1 | − 0.0759·2-s − 1.47·3-s − 0.994·4-s − 0.447·5-s + 0.111·6-s − 0.604·7-s + 0.151·8-s + 1.16·9-s + 0.0339·10-s − 0.301·11-s + 1.46·12-s + 0.353·13-s + 0.0458·14-s + 0.658·15-s + 0.982·16-s − 0.295·17-s − 0.0886·18-s + 0.229·19-s + 0.444·20-s + 0.889·21-s + 0.0228·22-s − 0.168·23-s − 0.222·24-s + 0.200·25-s − 0.0268·26-s − 0.246·27-s + 0.600·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.2161093838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2161093838\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 + 0.429T + 32T^{2} \) |
| 3 | \( 1 + 22.9T + 243T^{2} \) |
| 7 | \( 1 + 78.3T + 1.68e4T^{2} \) |
| 13 | \( 1 - 215.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 352.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 427.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 778.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.82e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.93e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.71e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.17e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.48e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.33e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.19e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.69e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.89e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.87e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.92e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.09e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.50e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.24e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.43e5T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.354734285626097056797696857543, −8.368763896381045186631312267643, −7.47879764640628393273673826551, −6.43532533768955275132844843066, −5.76736439011399003687845255240, −4.89396734523797928081701111475, −4.20039960544167630783152122634, −3.11795670498128335095187749143, −1.26829091578402388024785553077, −0.25089539531251105343273382999,
0.25089539531251105343273382999, 1.26829091578402388024785553077, 3.11795670498128335095187749143, 4.20039960544167630783152122634, 4.89396734523797928081701111475, 5.76736439011399003687845255240, 6.43532533768955275132844843066, 7.47879764640628393273673826551, 8.368763896381045186631312267643, 9.354734285626097056797696857543
