| L(s) = 1 | − 2·2-s + 4·4-s − 5·5-s + 18.9·7-s − 8·8-s + 10·10-s + 59.9·11-s − 13·13-s − 37.9·14-s + 16·16-s − 46.5·17-s + 102.·19-s − 20·20-s − 119.·22-s + 4.52·23-s + 25·25-s + 26·26-s + 75.9·28-s + 64.4·29-s + 264.·31-s − 32·32-s + 93.1·34-s − 94.9·35-s + 397.·37-s − 205.·38-s + 40·40-s − 101.·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.02·7-s − 0.353·8-s + 0.316·10-s + 1.64·11-s − 0.277·13-s − 0.724·14-s + 0.250·16-s − 0.664·17-s + 1.24·19-s − 0.223·20-s − 1.16·22-s + 0.0409·23-s + 0.200·25-s + 0.196·26-s + 0.512·28-s + 0.412·29-s + 1.53·31-s − 0.176·32-s + 0.469·34-s − 0.458·35-s + 1.76·37-s − 0.878·38-s + 0.158·40-s − 0.388·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.837330513\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.837330513\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 13 | \( 1 + 13T \) |
| good | 7 | \( 1 - 18.9T + 343T^{2} \) |
| 11 | \( 1 - 59.9T + 1.33e3T^{2} \) |
| 17 | \( 1 + 46.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 102.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 4.52T + 1.21e4T^{2} \) |
| 29 | \( 1 - 64.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 264.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 397.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 101.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 498.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 424.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 119.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 38.3T + 2.05e5T^{2} \) |
| 61 | \( 1 - 409.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.01e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 735.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 108.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 908.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.09e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 145.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 839.T + 9.12e5T^{2} \) |
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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
−9.431928803115621062834732986528, −8.447441915292712639213330486719, −8.006755913535589187572828777933, −6.96962981609360572871928893670, −6.36509336738283436197874156343, −5.02934524250855976907649869468, −4.22197455610273789586061479535, −3.05130688182241339941656957202, −1.69715154244688280427516241313, −0.836602967999309145616636039497,
0.836602967999309145616636039497, 1.69715154244688280427516241313, 3.05130688182241339941656957202, 4.22197455610273789586061479535, 5.02934524250855976907649869468, 6.36509336738283436197874156343, 6.96962981609360572871928893670, 8.006755913535589187572828777933, 8.447441915292712639213330486719, 9.431928803115621062834732986528
