| L(s) = 1 | + 1.41·2-s − 0.874·5-s + 7-s − 2.82·8-s − 1.23·10-s − 1.74·11-s + 13-s + 1.41·14-s − 4.00·16-s + 0.333·17-s + 6.70·19-s − 2.47·22-s + 1.20·23-s − 4.23·25-s + 1.41·26-s + 2.62·29-s − 2.76·31-s + 0.472·34-s − 0.874·35-s − 9·37-s + 9.48·38-s + 2.47·40-s + 5.11·41-s − 0.472·43-s + ⋯ |
| L(s) = 1 | + 1.00·2-s − 0.390·5-s + 0.377·7-s − 0.999·8-s − 0.390·10-s − 0.527·11-s + 0.277·13-s + 0.377·14-s − 1.00·16-s + 0.0809·17-s + 1.53·19-s − 0.527·22-s + 0.251·23-s − 0.847·25-s + 0.277·26-s + 0.486·29-s − 0.496·31-s + 0.0809·34-s − 0.147·35-s − 1.47·37-s + 1.53·38-s + 0.390·40-s + 0.799·41-s − 0.0720·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 5 | \( 1 + 0.874T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 + 1.74T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 0.333T + 17T^{2} \) |
| 19 | \( 1 - 6.70T + 19T^{2} \) |
| 23 | \( 1 - 1.20T + 23T^{2} \) |
| 29 | \( 1 - 2.62T + 29T^{2} \) |
| 31 | \( 1 + 2.76T + 31T^{2} \) |
| 37 | \( 1 + 9T + 37T^{2} \) |
| 41 | \( 1 - 5.11T + 41T^{2} \) |
| 43 | \( 1 + 0.472T + 43T^{2} \) |
| 47 | \( 1 - 6.73T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 5.24T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 + 9T + 67T^{2} \) |
| 71 | \( 1 - 6.65T + 71T^{2} \) |
| 73 | \( 1 + 9.18T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 4.37T + 83T^{2} \) |
| 89 | \( 1 - 1.74T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−7.71455718286811048619278012172, −6.96873573798366082496423003812, −6.04337758940517804930360718875, −5.39332953874366518824432359423, −4.88957933942096154908765494048, −4.06603768459230271055279829165, −3.37982218025050319148785803597, −2.69425863184597450378396485430, −1.40030054106051471165267504356, 0,
1.40030054106051471165267504356, 2.69425863184597450378396485430, 3.37982218025050319148785803597, 4.06603768459230271055279829165, 4.88957933942096154908765494048, 5.39332953874366518824432359423, 6.04337758940517804930360718875, 6.96873573798366082496423003812, 7.71455718286811048619278012172
