| L(s) = 1 | + 3.54·5-s − 3.54·7-s + 0.858·11-s − 1.49·13-s + 2·17-s − 19-s − 5.76·23-s + 7.59·25-s − 7.23·29-s + 6.26·31-s − 12.5·35-s + 0.828·37-s + 1.96·41-s − 5.76·43-s − 4.90·47-s + 5.59·49-s + 0.807·53-s + 3.04·55-s − 5.04·59-s + 9.64·61-s − 5.31·65-s − 9.54·67-s − 3.78·71-s − 10.7·73-s − 3.04·77-s + 8.87·79-s − 9.40·83-s + ⋯ |
| L(s) = 1 | + 1.58·5-s − 1.34·7-s + 0.258·11-s − 0.415·13-s + 0.485·17-s − 0.229·19-s − 1.20·23-s + 1.51·25-s − 1.34·29-s + 1.12·31-s − 2.12·35-s + 0.136·37-s + 0.307·41-s − 0.879·43-s − 0.716·47-s + 0.799·49-s + 0.110·53-s + 0.410·55-s − 0.657·59-s + 1.23·61-s − 0.659·65-s − 1.16·67-s − 0.448·71-s − 1.25·73-s − 0.347·77-s + 0.998·79-s − 1.03·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 3.54T + 5T^{2} \) |
| 7 | \( 1 + 3.54T + 7T^{2} \) |
| 11 | \( 1 - 0.858T + 11T^{2} \) |
| 13 | \( 1 + 1.49T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 23 | \( 1 + 5.76T + 23T^{2} \) |
| 29 | \( 1 + 7.23T + 29T^{2} \) |
| 31 | \( 1 - 6.26T + 31T^{2} \) |
| 37 | \( 1 - 0.828T + 37T^{2} \) |
| 41 | \( 1 - 1.96T + 41T^{2} \) |
| 43 | \( 1 + 5.76T + 43T^{2} \) |
| 47 | \( 1 + 4.90T + 47T^{2} \) |
| 53 | \( 1 - 0.807T + 53T^{2} \) |
| 59 | \( 1 + 5.04T + 59T^{2} \) |
| 61 | \( 1 - 9.64T + 61T^{2} \) |
| 67 | \( 1 + 9.54T + 67T^{2} \) |
| 71 | \( 1 + 3.78T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 8.87T + 79T^{2} \) |
| 83 | \( 1 + 9.40T + 83T^{2} \) |
| 89 | \( 1 + 7.83T + 89T^{2} \) |
| 97 | \( 1 + 3.33T + 97T^{2} \) |
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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
−7.30182839144409713198686872040, −6.59395607979482772556078782374, −6.05082448609347951366374708652, −5.67070886491340644594611397253, −4.75173401409288168062401431385, −3.78176074959013651633187577073, −2.97395825304730223098958750266, −2.24957255296239727901227632741, −1.40613504192436554115014728728, 0,
1.40613504192436554115014728728, 2.24957255296239727901227632741, 2.97395825304730223098958750266, 3.78176074959013651633187577073, 4.75173401409288168062401431385, 5.67070886491340644594611397253, 6.05082448609347951366374708652, 6.59395607979482772556078782374, 7.30182839144409713198686872040
