| L(s) = 1 | + 2.79·2-s + 5.79·4-s − 3·5-s + 7-s + 10.5·8-s − 8.37·10-s + 11-s + 13-s + 2.79·14-s + 17.9·16-s + 1.58·17-s + 2.58·19-s − 17.3·20-s + 2.79·22-s − 3.58·23-s + 4·25-s + 2.79·26-s + 5.79·28-s − 10.1·29-s − 5.58·31-s + 28.9·32-s + 4.41·34-s − 3·35-s + 37-s + 7.20·38-s − 31.7·40-s − 7.16·41-s + ⋯ |
| L(s) = 1 | + 1.97·2-s + 2.89·4-s − 1.34·5-s + 0.377·7-s + 3.74·8-s − 2.64·10-s + 0.301·11-s + 0.277·13-s + 0.746·14-s + 4.48·16-s + 0.383·17-s + 0.592·19-s − 3.88·20-s + 0.595·22-s − 0.747·23-s + 0.800·25-s + 0.547·26-s + 1.09·28-s − 1.88·29-s − 1.00·31-s + 5.11·32-s + 0.757·34-s − 0.507·35-s + 0.164·37-s + 1.16·38-s − 5.01·40-s − 1.11·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.426293635\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.426293635\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2.79T + 2T^{2} \) |
| 5 | \( 1 + 3T + 5T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 1.58T + 17T^{2} \) |
| 19 | \( 1 - 2.58T + 19T^{2} \) |
| 23 | \( 1 + 3.58T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 + 5.58T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 + 7.16T + 41T^{2} \) |
| 43 | \( 1 + 7.58T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 - 0.417T + 53T^{2} \) |
| 59 | \( 1 - 4.58T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 0.582T + 67T^{2} \) |
| 71 | \( 1 - 7.16T + 71T^{2} \) |
| 73 | \( 1 - 7T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 - 2.41T + 83T^{2} \) |
| 89 | \( 1 - 9.16T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−11.20363639310005632484633214780, −9.966496820274017928775391258543, −8.251434300043529018213532833211, −7.53518203195715746265288408686, −6.81147182438356980663234845793, −5.67153197742313612986465513592, −4.88364377100089667363934438738, −3.79894712507237977365226115983, −3.45514576744159127624363870826, −1.82898168095775629493084590715,
1.82898168095775629493084590715, 3.45514576744159127624363870826, 3.79894712507237977365226115983, 4.88364377100089667363934438738, 5.67153197742313612986465513592, 6.81147182438356980663234845793, 7.53518203195715746265288408686, 8.251434300043529018213532833211, 9.966496820274017928775391258543, 11.20363639310005632484633214780
