| L(s) = 1 | + 5-s − 3.90·7-s − 0.468·11-s + 1.90·13-s + 1.10·17-s + 19-s − 2.17·23-s + 25-s + 0.796·29-s + 7.24·31-s − 3.90·35-s − 6.23·37-s − 8.40·41-s − 7.18·43-s + 2.17·47-s + 8.24·49-s + 13.5·53-s − 0.468·55-s − 11.9·59-s + 0.780·61-s + 1.90·65-s + 13.2·67-s − 14.3·71-s − 2.95·73-s + 1.82·77-s − 14.6·79-s + 11.2·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.47·7-s − 0.141·11-s + 0.528·13-s + 0.268·17-s + 0.229·19-s − 0.452·23-s + 0.200·25-s + 0.147·29-s + 1.30·31-s − 0.659·35-s − 1.02·37-s − 1.31·41-s − 1.09·43-s + 0.316·47-s + 1.17·49-s + 1.86·53-s − 0.0631·55-s − 1.55·59-s + 0.0999·61-s + 0.236·65-s + 1.61·67-s − 1.70·71-s − 0.345·73-s + 0.208·77-s − 1.65·79-s + 1.23·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 3.90T + 7T^{2} \) |
| 11 | \( 1 + 0.468T + 11T^{2} \) |
| 13 | \( 1 - 1.90T + 13T^{2} \) |
| 17 | \( 1 - 1.10T + 17T^{2} \) |
| 23 | \( 1 + 2.17T + 23T^{2} \) |
| 29 | \( 1 - 0.796T + 29T^{2} \) |
| 31 | \( 1 - 7.24T + 31T^{2} \) |
| 37 | \( 1 + 6.23T + 37T^{2} \) |
| 41 | \( 1 + 8.40T + 41T^{2} \) |
| 43 | \( 1 + 7.18T + 43T^{2} \) |
| 47 | \( 1 - 2.17T + 47T^{2} \) |
| 53 | \( 1 - 13.5T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 - 0.780T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 + 14.3T + 71T^{2} \) |
| 73 | \( 1 + 2.95T + 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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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
−7.53001030522685103883923390463, −6.72718615875197669438091773649, −6.31503511905559904636431788104, −5.62205565703813515738413152878, −4.83662007157765174790773123120, −3.77031211946358049397173134723, −3.21856053539771664144718154269, −2.40108619857595633889353966534, −1.25457106469569294719990132059, 0,
1.25457106469569294719990132059, 2.40108619857595633889353966534, 3.21856053539771664144718154269, 3.77031211946358049397173134723, 4.83662007157765174790773123120, 5.62205565703813515738413152878, 6.31503511905559904636431788104, 6.72718615875197669438091773649, 7.53001030522685103883923390463
