| L(s) = 1 | − 2-s − 0.970·3-s + 4-s − 5-s + 0.970·6-s − 0.827·7-s − 8-s − 2.05·9-s + 10-s − 11-s − 0.970·12-s − 0.399·13-s + 0.827·14-s + 0.970·15-s + 16-s − 2.38·17-s + 2.05·18-s − 3.72·19-s − 20-s + 0.802·21-s + 22-s + 4.45·23-s + 0.970·24-s + 25-s + 0.399·26-s + 4.90·27-s − 0.827·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.560·3-s + 0.5·4-s − 0.447·5-s + 0.396·6-s − 0.312·7-s − 0.353·8-s − 0.686·9-s + 0.316·10-s − 0.301·11-s − 0.280·12-s − 0.110·13-s + 0.221·14-s + 0.250·15-s + 0.250·16-s − 0.577·17-s + 0.485·18-s − 0.854·19-s − 0.223·20-s + 0.175·21-s + 0.213·22-s + 0.928·23-s + 0.198·24-s + 0.200·25-s + 0.0784·26-s + 0.944·27-s − 0.156·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 + 0.970T + 3T^{2} \) |
| 7 | \( 1 + 0.827T + 7T^{2} \) |
| 13 | \( 1 + 0.399T + 13T^{2} \) |
| 17 | \( 1 + 2.38T + 17T^{2} \) |
| 19 | \( 1 + 3.72T + 19T^{2} \) |
| 23 | \( 1 - 4.45T + 23T^{2} \) |
| 29 | \( 1 - 3.28T + 29T^{2} \) |
| 31 | \( 1 - 8.18T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 - 3.28T + 41T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 - 8.41T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 4.03T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 4.77T + 73T^{2} \) |
| 79 | \( 1 + 0.730T + 79T^{2} \) |
| 83 | \( 1 - 6.10T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 + 5.35T + 97T^{2} \) |
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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
−8.006964253032105088282289419686, −7.29118185009065996036678633038, −6.34633574962309406411622236619, −6.11932533409728687378317386365, −4.94793643803363717526117875748, −4.32270038298040755248359948900, −3.03836204451721399019801273578, −2.49252415750512014592366296811, −1.00073961212350457795142055420, 0,
1.00073961212350457795142055420, 2.49252415750512014592366296811, 3.03836204451721399019801273578, 4.32270038298040755248359948900, 4.94793643803363717526117875748, 6.11932533409728687378317386365, 6.34633574962309406411622236619, 7.29118185009065996036678633038, 8.006964253032105088282289419686
