L(s) = 1 | + 4.11·2-s + 8.95·4-s − 5·5-s − 20.0·7-s + 3.92·8-s − 20.5·10-s + 1.67·11-s + 11.1·13-s − 82.4·14-s − 55.4·16-s − 8.98·17-s − 50.6·19-s − 44.7·20-s + 6.90·22-s − 214.·23-s + 25·25-s + 45.7·26-s − 179.·28-s + 76.9·29-s − 273.·31-s − 259.·32-s − 37.0·34-s + 100.·35-s − 137.·37-s − 208.·38-s − 19.6·40-s + 53.3·41-s + ⋯ |
L(s) = 1 | + 1.45·2-s + 1.11·4-s − 0.447·5-s − 1.08·7-s + 0.173·8-s − 0.651·10-s + 0.0460·11-s + 0.237·13-s − 1.57·14-s − 0.866·16-s − 0.128·17-s − 0.611·19-s − 0.500·20-s + 0.0669·22-s − 1.94·23-s + 0.200·25-s + 0.345·26-s − 1.20·28-s + 0.492·29-s − 1.58·31-s − 1.43·32-s − 0.186·34-s + 0.483·35-s − 0.609·37-s − 0.890·38-s − 0.0775·40-s + 0.203·41-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 | \( 1 - 4.11T + 8T^{2} \) |
| 7 | \( 1 + 20.0T + 343T^{2} \) |
| 11 | \( 1 - 1.67T + 1.33e3T^{2} \) |
| 13 | \( 1 - 11.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 8.98T + 4.91e3T^{2} \) |
| 19 | \( 1 + 50.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 214.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 76.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 273.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 137.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 53.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 295.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 194.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 450.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 481.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 675.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 894.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 721.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 915.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 59.6T + 4.93e5T^{2} \) |
| 83 | \( 1 + 742.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.54e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.12e3T + 9.12e5T^{2} \) |
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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
−10.62799160630676598679751401090, −9.548688207949698005993560538694, −8.495199926859801469288900913062, −7.17954646745873978601620621167, −6.29566170091375281375861737049, −5.52990790476410487181551508736, −4.16866446237938234598204489329, −3.60635695483203289651526544597, −2.35426089042293201409759156858, 0,
2.35426089042293201409759156858, 3.60635695483203289651526544597, 4.16866446237938234598204489329, 5.52990790476410487181551508736, 6.29566170091375281375861737049, 7.17954646745873978601620621167, 8.495199926859801469288900913062, 9.548688207949698005993560538694, 10.62799160630676598679751401090