| L(s) = 1 | − 5.42·2-s + 21.4·4-s + 5·5-s − 10.2·7-s − 73.0·8-s − 27.1·10-s − 12.6·11-s + 84.2·13-s + 55.5·14-s + 224.·16-s − 6.89·17-s − 79.7·19-s + 107.·20-s + 68.5·22-s + 73.8·23-s + 25·25-s − 457.·26-s − 219.·28-s − 29·29-s + 0.254·31-s − 635.·32-s + 37.4·34-s − 51.1·35-s + 40.4·37-s + 432.·38-s − 365.·40-s − 208.·41-s + ⋯ |
| L(s) = 1 | − 1.91·2-s + 2.68·4-s + 0.447·5-s − 0.552·7-s − 3.22·8-s − 0.858·10-s − 0.346·11-s + 1.79·13-s + 1.06·14-s + 3.51·16-s − 0.0983·17-s − 0.962·19-s + 1.19·20-s + 0.664·22-s + 0.669·23-s + 0.200·25-s − 3.44·26-s − 1.48·28-s − 0.185·29-s + 0.00147·31-s − 3.50·32-s + 0.188·34-s − 0.247·35-s + 0.179·37-s + 1.84·38-s − 1.44·40-s − 0.793·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8257481923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8257481923\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 + 5.42T + 8T^{2} \) |
| 7 | \( 1 + 10.2T + 343T^{2} \) |
| 11 | \( 1 + 12.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 84.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 6.89T + 4.91e3T^{2} \) |
| 19 | \( 1 + 79.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 73.8T + 1.21e4T^{2} \) |
| 31 | \( 1 - 0.254T + 2.97e4T^{2} \) |
| 37 | \( 1 - 40.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 208.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 194.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 522.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 294.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 196.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 189.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 570.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.12e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 581.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 255.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 983.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 678.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 30.8T + 9.12e5T^{2} \) |
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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
−9.168325601738725054973203278229, −8.614018995695859282453440683987, −7.988602201365386443516327851363, −6.84606818260857607075460626778, −6.41507176708109832898151333477, −5.54356370076393828946485483322, −3.70167586759197876619678908477, −2.64022643579589217851844746362, −1.62451858496124770010980823076, −0.62430851322047967491468507308,
0.62430851322047967491468507308, 1.62451858496124770010980823076, 2.64022643579589217851844746362, 3.70167586759197876619678908477, 5.54356370076393828946485483322, 6.41507176708109832898151333477, 6.84606818260857607075460626778, 7.988602201365386443516327851363, 8.614018995695859282453440683987, 9.168325601738725054973203278229
