| L(s) = 1 | + 4.46·2-s + 11.9·4-s + 6.53·5-s + 9.32·7-s + 17.5·8-s + 29.1·10-s − 11·11-s + 1.89·13-s + 41.6·14-s − 17.1·16-s + 61.2·17-s − 121.·19-s + 77.9·20-s − 49.1·22-s + 117.·23-s − 82.2·25-s + 8.45·26-s + 111.·28-s − 139.·29-s − 314.·31-s − 216.·32-s + 273.·34-s + 60.9·35-s + 88.6·37-s − 544.·38-s + 114.·40-s + 92.1·41-s + ⋯ |
| L(s) = 1 | + 1.57·2-s + 1.49·4-s + 0.584·5-s + 0.503·7-s + 0.774·8-s + 0.922·10-s − 0.301·11-s + 0.0404·13-s + 0.794·14-s − 0.267·16-s + 0.873·17-s − 1.47·19-s + 0.871·20-s − 0.475·22-s + 1.06·23-s − 0.658·25-s + 0.0638·26-s + 0.750·28-s − 0.896·29-s − 1.81·31-s − 1.19·32-s + 1.37·34-s + 0.294·35-s + 0.393·37-s − 2.32·38-s + 0.453·40-s + 0.350·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.729743763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.729743763\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 4.46T + 8T^{2} \) |
| 5 | \( 1 - 6.53T + 125T^{2} \) |
| 7 | \( 1 - 9.32T + 343T^{2} \) |
| 13 | \( 1 - 1.89T + 2.19e3T^{2} \) |
| 17 | \( 1 - 61.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 121.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 117.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 139.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 314.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 88.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 92.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 396.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 82.7T + 1.03e5T^{2} \) |
| 53 | \( 1 - 581.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 697.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 184.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 219.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 886.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 338.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 208.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 463.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.13e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.52e3T + 9.12e5T^{2} \) |
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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
−13.26580437011742040174854867611, −12.78392508566684668116245269604, −11.54489047071759861418940900331, −10.59370404535768446421974108556, −9.058810028364249598034084213429, −7.43126313196315315678457759983, −6.03553423603517429178052398660, −5.16594615695106684174194864653, −3.82351732570749924444722014957, −2.19646146003414190013375980139,
2.19646146003414190013375980139, 3.82351732570749924444722014957, 5.16594615695106684174194864653, 6.03553423603517429178052398660, 7.43126313196315315678457759983, 9.058810028364249598034084213429, 10.59370404535768446421974108556, 11.54489047071759861418940900331, 12.78392508566684668116245269604, 13.26580437011742040174854867611
