| L(s) = 1 | + 3.07·2-s + 16.4·3-s − 22.5·4-s + 50.4·6-s + 104.·7-s − 167.·8-s + 26.0·9-s − 730.·11-s − 369.·12-s + 707.·13-s + 322.·14-s + 206.·16-s − 389.·17-s + 80.1·18-s − 973.·19-s + 1.72e3·21-s − 2.24e3·22-s + 1.73e3·23-s − 2.75e3·24-s + 2.17e3·26-s − 3.55e3·27-s − 2.36e3·28-s + 7.77e3·29-s − 4.60e3·31-s + 5.99e3·32-s − 1.19e4·33-s − 1.19e3·34-s + ⋯ |
| L(s) = 1 | + 0.543·2-s + 1.05·3-s − 0.704·4-s + 0.571·6-s + 0.809·7-s − 0.926·8-s + 0.107·9-s − 1.82·11-s − 0.741·12-s + 1.16·13-s + 0.439·14-s + 0.201·16-s − 0.326·17-s + 0.0582·18-s − 0.618·19-s + 0.851·21-s − 0.988·22-s + 0.685·23-s − 0.974·24-s + 0.630·26-s − 0.939·27-s − 0.570·28-s + 1.71·29-s − 0.859·31-s + 1.03·32-s − 1.91·33-s − 0.177·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.140061172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.140061172\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 + 1.84e3T \) |
| good | 2 | \( 1 - 3.07T + 32T^{2} \) |
| 3 | \( 1 - 16.4T + 243T^{2} \) |
| 7 | \( 1 - 104.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 730.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 707.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 389.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 973.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.73e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.77e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.60e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.49e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.41e4T + 1.15e8T^{2} \) |
| 47 | \( 1 + 6.95e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.15e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.39e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.58e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.79e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.10e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.22e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.05e5T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.44e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.32e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.46e5T + 8.58e9T^{2} \) |
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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
−8.866909724151055133521701545509, −8.265156381727010752719834807020, −7.998676170495318358296583779499, −6.56810012181365427632639866609, −5.42346146320458198346941923706, −4.85414420356462845070180260667, −3.81830936468125278671649543145, −2.97687782286073730709109359340, −2.13077426300263557291217000669, −0.65169679879549968883624378490,
0.65169679879549968883624378490, 2.13077426300263557291217000669, 2.97687782286073730709109359340, 3.81830936468125278671649543145, 4.85414420356462845070180260667, 5.42346146320458198346941923706, 6.56810012181365427632639866609, 7.998676170495318358296583779499, 8.265156381727010752719834807020, 8.866909724151055133521701545509
