| L(s) = 1 | − 17.5·3-s + 87.4·5-s − 76.9·7-s + 64.9·9-s + 99.2·11-s − 79.5·13-s − 1.53e3·15-s − 465.·17-s + 361·19-s + 1.34e3·21-s − 2.23e3·23-s + 4.52e3·25-s + 3.12e3·27-s + 757.·29-s − 2.71e3·31-s − 1.74e3·33-s − 6.72e3·35-s + 1.03e4·37-s + 1.39e3·39-s − 6.94e3·41-s + 2.10e3·43-s + 5.68e3·45-s + 2.05e4·47-s − 1.08e4·49-s + 8.17e3·51-s − 1.75e4·53-s + 8.67e3·55-s + ⋯ |
| L(s) = 1 | − 1.12·3-s + 1.56·5-s − 0.593·7-s + 0.267·9-s + 0.247·11-s − 0.130·13-s − 1.76·15-s − 0.391·17-s + 0.229·19-s + 0.668·21-s − 0.879·23-s + 1.44·25-s + 0.824·27-s + 0.167·29-s − 0.507·31-s − 0.278·33-s − 0.928·35-s + 1.23·37-s + 0.146·39-s − 0.645·41-s + 0.173·43-s + 0.418·45-s + 1.35·47-s − 0.647·49-s + 0.440·51-s − 0.859·53-s + 0.386·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 361T \) |
| good | 3 | \( 1 + 17.5T + 243T^{2} \) |
| 5 | \( 1 - 87.4T + 3.12e3T^{2} \) |
| 7 | \( 1 + 76.9T + 1.68e4T^{2} \) |
| 11 | \( 1 - 99.2T + 1.61e5T^{2} \) |
| 13 | \( 1 + 79.5T + 3.71e5T^{2} \) |
| 17 | \( 1 + 465.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 2.23e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 757.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.71e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.03e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.94e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.10e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.05e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.75e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.45e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.09e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.82e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.88e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.36e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.90e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.94e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.89e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.74e4T + 8.58e9T^{2} \) |
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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
−10.33557774733212536503516869527, −9.715153058743073032028644600024, −8.774571576979415285386674229143, −7.12523285712862573177109019062, −6.05254985928792593710792077297, −5.80735219101509359118934089195, −4.54472918594237244578286934520, −2.77853606508981614048291090238, −1.45333017393190195132315931897, 0,
1.45333017393190195132315931897, 2.77853606508981614048291090238, 4.54472918594237244578286934520, 5.80735219101509359118934089195, 6.05254985928792593710792077297, 7.12523285712862573177109019062, 8.774571576979415285386674229143, 9.715153058743073032028644600024, 10.33557774733212536503516869527
