| L(s) = 1 | − 0.174·2-s − 7.96·4-s + 5·5-s + 8.46·7-s + 2.79·8-s − 0.874·10-s − 31.5·11-s + 26.8·13-s − 1.48·14-s + 63.2·16-s − 44.3·17-s − 90.2·19-s − 39.8·20-s + 5.51·22-s + 194.·23-s + 25·25-s − 4.70·26-s − 67.4·28-s + 3.74·29-s − 251.·31-s − 33.4·32-s + 7.75·34-s + 42.3·35-s − 62.2·37-s + 15.7·38-s + 13.9·40-s − 204.·41-s + ⋯ |
| L(s) = 1 | − 0.0618·2-s − 0.996·4-s + 0.447·5-s + 0.456·7-s + 0.123·8-s − 0.0276·10-s − 0.863·11-s + 0.573·13-s − 0.0282·14-s + 0.988·16-s − 0.632·17-s − 1.08·19-s − 0.445·20-s + 0.0534·22-s + 1.76·23-s + 0.200·25-s − 0.0354·26-s − 0.455·28-s + 0.0239·29-s − 1.45·31-s − 0.184·32-s + 0.0391·34-s + 0.204·35-s − 0.276·37-s + 0.0673·38-s + 0.0552·40-s − 0.778·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 2 | \( 1 + 0.174T + 8T^{2} \) |
| 7 | \( 1 - 8.46T + 343T^{2} \) |
| 11 | \( 1 + 31.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 26.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 44.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 90.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 194.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 3.74T + 2.43e4T^{2} \) |
| 31 | \( 1 + 251.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 62.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 204.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 527.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 155.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 141.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 493.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 759.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 543.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 928.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 608.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 614.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.07e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.50e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 332.T + 9.12e5T^{2} \) |
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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
−10.51485529132364293518212862134, −9.263915550416091765521365810651, −8.705064356640272361350817850249, −7.78198917260965230225193949356, −6.51936857598590904718316523056, −5.30095883628344485014299751185, −4.63155593026877467019211178082, −3.26282067515033001679207004643, −1.65679112050516327814094621851, 0,
1.65679112050516327814094621851, 3.26282067515033001679207004643, 4.63155593026877467019211178082, 5.30095883628344485014299751185, 6.51936857598590904718316523056, 7.78198917260965230225193949356, 8.705064356640272361350817850249, 9.263915550416091765521365810651, 10.51485529132364293518212862134
