| L(s) = 1 | + 2.33·5-s + 2.33·7-s + 0.771·11-s − 6.14·13-s − 2·17-s − 19-s − 8.02·23-s + 0.466·25-s + 6.38·29-s − 2.88·31-s + 5.46·35-s + 7.49·37-s − 10.2·41-s + 8.02·43-s − 7.25·47-s − 1.53·49-s − 9.70·53-s + 1.80·55-s + 3.80·59-s − 8.01·61-s − 14.3·65-s − 12.9·67-s + 11.6·71-s − 13.5·73-s + 1.80·77-s + 14.2·79-s + 11.1·83-s + ⋯ |
| L(s) = 1 | + 1.04·5-s + 0.883·7-s + 0.232·11-s − 1.70·13-s − 0.485·17-s − 0.229·19-s − 1.67·23-s + 0.0933·25-s + 1.18·29-s − 0.517·31-s + 0.924·35-s + 1.23·37-s − 1.60·41-s + 1.22·43-s − 1.05·47-s − 0.219·49-s − 1.33·53-s + 0.243·55-s + 0.495·59-s − 1.02·61-s − 1.78·65-s − 1.58·67-s + 1.38·71-s − 1.58·73-s + 0.205·77-s + 1.60·79-s + 1.22·83-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 | \( 1 - 2.33T + 5T^{2} \) |
| 7 | \( 1 - 2.33T + 7T^{2} \) |
| 11 | \( 1 - 0.771T + 11T^{2} \) |
| 13 | \( 1 + 6.14T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 23 | \( 1 + 8.02T + 23T^{2} \) |
| 29 | \( 1 - 6.38T + 29T^{2} \) |
| 31 | \( 1 + 2.88T + 31T^{2} \) |
| 37 | \( 1 - 7.49T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 - 8.02T + 43T^{2} \) |
| 47 | \( 1 + 7.25T + 47T^{2} \) |
| 53 | \( 1 + 9.70T + 53T^{2} \) |
| 59 | \( 1 - 3.80T + 59T^{2} \) |
| 61 | \( 1 + 8.01T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 + 0.151T + 89T^{2} \) |
| 97 | \( 1 + 5.34T + 97T^{2} \) |
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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
−7.66667394602552220143747648795, −6.59932049794746064380449041254, −6.18856559051557454527031648351, −5.27246375345138347480188479402, −4.76921006733939512692655882350, −4.09298908709797765048841564578, −2.82688661399930669177662289083, −2.13105777701585075044408233620, −1.53781235065287960630858154652, 0,
1.53781235065287960630858154652, 2.13105777701585075044408233620, 2.82688661399930669177662289083, 4.09298908709797765048841564578, 4.76921006733939512692655882350, 5.27246375345138347480188479402, 6.18856559051557454527031648351, 6.59932049794746064380449041254, 7.66667394602552220143747648795
