| L(s) = 1 | + 3.02·2-s − 6.47·3-s + 1.16·4-s − 14.0·5-s − 19.6·6-s − 20.6·8-s + 14.9·9-s − 42.5·10-s + 1.79·11-s − 7.57·12-s − 31.5·13-s + 90.9·15-s − 71.9·16-s + 17.8·17-s + 45.2·18-s − 34.8·19-s − 16.4·20-s + 5.43·22-s + 199.·23-s + 133.·24-s + 72.0·25-s − 95.3·26-s + 78.0·27-s + 275.·29-s + 275.·30-s − 116.·31-s − 52.4·32-s + ⋯ |
| L(s) = 1 | + 1.07·2-s − 1.24·3-s + 0.146·4-s − 1.25·5-s − 1.33·6-s − 0.914·8-s + 0.553·9-s − 1.34·10-s + 0.0491·11-s − 0.182·12-s − 0.672·13-s + 1.56·15-s − 1.12·16-s + 0.253·17-s + 0.592·18-s − 0.420·19-s − 0.183·20-s + 0.0526·22-s + 1.80·23-s + 1.13·24-s + 0.576·25-s − 0.719·26-s + 0.556·27-s + 1.76·29-s + 1.67·30-s − 0.675·31-s − 0.289·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 3.02T + 8T^{2} \) |
| 3 | \( 1 + 6.47T + 27T^{2} \) |
| 5 | \( 1 + 14.0T + 125T^{2} \) |
| 11 | \( 1 - 1.79T + 1.33e3T^{2} \) |
| 13 | \( 1 + 31.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 17.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 34.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 199.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 275.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 116.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 32.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 326.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 450.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 364.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 410.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 836.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 7.21T + 2.26e5T^{2} \) |
| 67 | \( 1 - 671.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 717.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 548.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 195.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.28e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 607.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 665.T + 9.12e5T^{2} \) |
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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
−8.235212897300102319375083059607, −6.88967189989457340948301064117, −6.79177034462361393874467911482, −5.50156543581799994577707318556, −5.08198855972872823230257362656, −4.40187075234642897200203317108, −3.58502154930059754993688885663, −2.70481513809035475055630359156, −0.841958342395072087663290047946, 0,
0.841958342395072087663290047946, 2.70481513809035475055630359156, 3.58502154930059754993688885663, 4.40187075234642897200203317108, 5.08198855972872823230257362656, 5.50156543581799994577707318556, 6.79177034462361393874467911482, 6.88967189989457340948301064117, 8.235212897300102319375083059607
