| L(s) = 1 | − 3.26·3-s + 3.04·7-s + 7.66·9-s − 1.59·11-s + 1.63·13-s − 3.25·17-s − 6.90·19-s − 9.95·21-s + 2.58·23-s − 15.2·27-s − 8.93·29-s + 4.16·31-s + 5.22·33-s − 0.951·37-s − 5.34·39-s − 2.26·41-s + 3.14·43-s + 1.59·47-s + 2.29·49-s + 10.6·51-s + 4.40·53-s + 22.5·57-s − 5.33·59-s + 6.87·61-s + 23.3·63-s − 1.73·67-s − 8.45·69-s + ⋯ |
| L(s) = 1 | − 1.88·3-s + 1.15·7-s + 2.55·9-s − 0.482·11-s + 0.453·13-s − 0.789·17-s − 1.58·19-s − 2.17·21-s + 0.539·23-s − 2.93·27-s − 1.65·29-s + 0.748·31-s + 0.909·33-s − 0.156·37-s − 0.855·39-s − 0.353·41-s + 0.480·43-s + 0.233·47-s + 0.327·49-s + 1.48·51-s + 0.604·53-s + 2.98·57-s − 0.694·59-s + 0.879·61-s + 2.94·63-s − 0.212·67-s − 1.01·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8300450244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8300450244\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 3.26T + 3T^{2} \) |
| 7 | \( 1 - 3.04T + 7T^{2} \) |
| 11 | \( 1 + 1.59T + 11T^{2} \) |
| 13 | \( 1 - 1.63T + 13T^{2} \) |
| 17 | \( 1 + 3.25T + 17T^{2} \) |
| 19 | \( 1 + 6.90T + 19T^{2} \) |
| 23 | \( 1 - 2.58T + 23T^{2} \) |
| 29 | \( 1 + 8.93T + 29T^{2} \) |
| 31 | \( 1 - 4.16T + 31T^{2} \) |
| 37 | \( 1 + 0.951T + 37T^{2} \) |
| 41 | \( 1 + 2.26T + 41T^{2} \) |
| 43 | \( 1 - 3.14T + 43T^{2} \) |
| 47 | \( 1 - 1.59T + 47T^{2} \) |
| 53 | \( 1 - 4.40T + 53T^{2} \) |
| 59 | \( 1 + 5.33T + 59T^{2} \) |
| 61 | \( 1 - 6.87T + 61T^{2} \) |
| 67 | \( 1 + 1.73T + 67T^{2} \) |
| 71 | \( 1 - 5.38T + 71T^{2} \) |
| 73 | \( 1 + 4.32T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 - 8.19T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−7.41969488217850701349362299604, −6.89038068773575688773032201507, −6.12365608178732825825828232441, −5.68312909014009289217558142477, −4.87479028939351673258390371211, −4.53156466543831029537088841845, −3.78430725021108764056311064527, −2.23554225672186874654238377820, −1.54707306874606605502381422756, −0.49267443721629063856659694845,
0.49267443721629063856659694845, 1.54707306874606605502381422756, 2.23554225672186874654238377820, 3.78430725021108764056311064527, 4.53156466543831029537088841845, 4.87479028939351673258390371211, 5.68312909014009289217558142477, 6.12365608178732825825828232441, 6.89038068773575688773032201507, 7.41969488217850701349362299604
