| L(s) = 1 | − 2-s + 4-s − 3.16·7-s − 8-s + 2.16·11-s − 1.16·13-s + 3.16·14-s + 16-s − 7.16·17-s − 19-s − 2.16·22-s − 7.32·23-s + 1.16·26-s − 3.16·28-s + 10.1·29-s − 7.32·31-s − 32-s + 7.16·34-s + 10·37-s + 38-s − 6.32·41-s − 2.83·43-s + 2.16·44-s + 7.32·46-s − 6·47-s + 3.00·49-s − 1.16·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.19·7-s − 0.353·8-s + 0.651·11-s − 0.322·13-s + 0.845·14-s + 0.250·16-s − 1.73·17-s − 0.229·19-s − 0.460·22-s − 1.52·23-s + 0.227·26-s − 0.597·28-s + 1.88·29-s − 1.31·31-s − 0.176·32-s + 1.22·34-s + 1.64·37-s + 0.162·38-s − 0.987·41-s − 0.432·43-s + 0.325·44-s + 1.07·46-s − 0.875·47-s + 0.428·49-s − 0.161·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6044867847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6044867847\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 3.16T + 7T^{2} \) |
| 11 | \( 1 - 2.16T + 11T^{2} \) |
| 13 | \( 1 + 1.16T + 13T^{2} \) |
| 17 | \( 1 + 7.16T + 17T^{2} \) |
| 23 | \( 1 + 7.32T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 + 7.32T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + 6.32T + 41T^{2} \) |
| 43 | \( 1 + 2.83T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 8.16T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 2.16T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 5.16T + 71T^{2} \) |
| 73 | \( 1 + T + 73T^{2} \) |
| 79 | \( 1 - 7.32T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 + 5.32T + 89T^{2} \) |
| 97 | \( 1 - 19.4T + 97T^{2} \) |
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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
−7.938211478461632835787986255338, −6.94329357498887064388330507808, −6.41067642359163845304717669472, −6.21438757615137251831359660253, −4.95371445025594578284804507689, −4.16883847927963709207677397466, −3.38485342064251915314969414336, −2.51694515807185686964469623713, −1.75772754313986829444406221493, −0.40483151281131082281971178295,
0.40483151281131082281971178295, 1.75772754313986829444406221493, 2.51694515807185686964469623713, 3.38485342064251915314969414336, 4.16883847927963709207677397466, 4.95371445025594578284804507689, 6.21438757615137251831359660253, 6.41067642359163845304717669472, 6.94329357498887064388330507808, 7.938211478461632835787986255338
