| L(s) = 1 | − 2·2-s + 4·4-s + 6.18·5-s + 25.7·7-s − 8·8-s − 12.3·10-s + 30.7·13-s − 51.4·14-s + 16·16-s − 47.9·17-s − 46.7·19-s + 24.7·20-s − 2.04·23-s − 86.7·25-s − 61.5·26-s + 102.·28-s − 59.0·29-s + 304.·31-s − 32·32-s + 95.9·34-s + 159.·35-s − 303.·37-s + 93.4·38-s − 49.4·40-s − 145.·41-s − 284.·43-s + 4.08·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.553·5-s + 1.38·7-s − 0.353·8-s − 0.391·10-s + 0.656·13-s − 0.981·14-s + 0.250·16-s − 0.684·17-s − 0.564·19-s + 0.276·20-s − 0.0185·23-s − 0.693·25-s − 0.464·26-s + 0.694·28-s − 0.377·29-s + 1.76·31-s − 0.176·32-s + 0.484·34-s + 0.768·35-s − 1.34·37-s + 0.399·38-s − 0.195·40-s − 0.554·41-s − 1.00·43-s + 0.0131·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 6.18T + 125T^{2} \) |
| 7 | \( 1 - 25.7T + 343T^{2} \) |
| 13 | \( 1 - 30.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 47.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 46.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 2.04T + 1.21e4T^{2} \) |
| 29 | \( 1 + 59.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 304.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 303.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 145.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 284.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 577.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 655.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 23.1T + 2.05e5T^{2} \) |
| 61 | \( 1 + 631.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 655.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 528.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 447.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 985.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 128.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.42e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 142.T + 9.12e5T^{2} \) |
| 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
−8.307025828237981956318660278106, −7.87389890658920936884407934341, −6.73947516012261210248154851713, −6.15597952401815853154636196106, −5.12553417237832862525395729672, −4.40297850189619770213841651274, −3.13832287133698409318842007126, −1.88988053026649869168680314098, −1.48485363045770181211517285788, 0,
1.48485363045770181211517285788, 1.88988053026649869168680314098, 3.13832287133698409318842007126, 4.40297850189619770213841651274, 5.12553417237832862525395729672, 6.15597952401815853154636196106, 6.73947516012261210248154851713, 7.87389890658920936884407934341, 8.307025828237981956318660278106
