| L(s) = 1 | − 2.37·2-s − 13.1·3-s − 26.3·4-s − 25·5-s + 31.1·6-s + 161.·7-s + 138.·8-s − 70.8·9-s + 59.3·10-s + 121·11-s + 346.·12-s − 854.·13-s − 382.·14-s + 328.·15-s + 515.·16-s − 773.·17-s + 168.·18-s − 361·19-s + 659.·20-s − 2.11e3·21-s − 287.·22-s + 3.01e3·23-s − 1.81e3·24-s + 625·25-s + 2.02e3·26-s + 4.11e3·27-s − 4.25e3·28-s + ⋯ |
| L(s) = 1 | − 0.419·2-s − 0.841·3-s − 0.824·4-s − 0.447·5-s + 0.353·6-s + 1.24·7-s + 0.765·8-s − 0.291·9-s + 0.187·10-s + 0.301·11-s + 0.693·12-s − 1.40·13-s − 0.521·14-s + 0.376·15-s + 0.503·16-s − 0.649·17-s + 0.122·18-s − 0.229·19-s + 0.368·20-s − 1.04·21-s − 0.126·22-s + 1.18·23-s − 0.644·24-s + 0.200·25-s + 0.588·26-s + 1.08·27-s − 1.02·28-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
| \(L(3)\) |
\(\approx\) |
\(0.6112389251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6112389251\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 2.37T + 32T^{2} \) |
| 3 | \( 1 + 13.1T + 243T^{2} \) |
| 7 | \( 1 - 161.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 854.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 773.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 3.01e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.56e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.62e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.17e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.71e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.64e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 7.28e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.28e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.55e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.67e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.69e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.65e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.75e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.89e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.78e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.07e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.70e4T + 8.58e9T^{2} \) |
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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
−9.047531692603974327650117772174, −8.433688168086369882067833808998, −7.64782867878220488044628004609, −6.80166301680677178872476709809, −5.54595140297386974115610803952, −4.73920573571355507503460050381, −4.42805833129723619001380572185, −2.80009927041868259029228712085, −1.39311553052216795780125815546, −0.41338659402449053437089015317,
0.41338659402449053437089015317, 1.39311553052216795780125815546, 2.80009927041868259029228712085, 4.42805833129723619001380572185, 4.73920573571355507503460050381, 5.54595140297386974115610803952, 6.80166301680677178872476709809, 7.64782867878220488044628004609, 8.433688168086369882067833808998, 9.047531692603974327650117772174
