L(s) = 1 | + 2.22·2-s + 2.97·4-s − 3.25·5-s − 0.940·7-s + 2.16·8-s − 7.26·10-s + 11-s + 1.71·13-s − 2.09·14-s − 1.11·16-s + 0.0942·17-s + 3.20·19-s − 9.67·20-s + 2.22·22-s + 6.07·23-s + 5.60·25-s + 3.83·26-s − 2.79·28-s + 3.73·29-s − 9.85·31-s − 6.81·32-s + 0.210·34-s + 3.06·35-s − 8.79·37-s + 7.15·38-s − 7.04·40-s − 3.45·41-s + ⋯ |
L(s) = 1 | + 1.57·2-s + 1.48·4-s − 1.45·5-s − 0.355·7-s + 0.764·8-s − 2.29·10-s + 0.301·11-s + 0.476·13-s − 0.560·14-s − 0.279·16-s + 0.0228·17-s + 0.736·19-s − 2.16·20-s + 0.475·22-s + 1.26·23-s + 1.12·25-s + 0.751·26-s − 0.527·28-s + 0.694·29-s − 1.76·31-s − 1.20·32-s + 0.0360·34-s + 0.517·35-s − 1.44·37-s + 1.16·38-s − 1.11·40-s − 0.539·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
good | 2 | \( 1 - 2.22T + 2T^{2} \) |
| 5 | \( 1 + 3.25T + 5T^{2} \) |
| 7 | \( 1 + 0.940T + 7T^{2} \) |
| 13 | \( 1 - 1.71T + 13T^{2} \) |
| 17 | \( 1 - 0.0942T + 17T^{2} \) |
| 19 | \( 1 - 3.20T + 19T^{2} \) |
| 23 | \( 1 - 6.07T + 23T^{2} \) |
| 29 | \( 1 - 3.73T + 29T^{2} \) |
| 31 | \( 1 + 9.85T + 31T^{2} \) |
| 37 | \( 1 + 8.79T + 37T^{2} \) |
| 41 | \( 1 + 3.45T + 41T^{2} \) |
| 43 | \( 1 + 9.13T + 43T^{2} \) |
| 47 | \( 1 - 4.55T + 47T^{2} \) |
| 53 | \( 1 + 2.28T + 53T^{2} \) |
| 59 | \( 1 + 0.677T + 59T^{2} \) |
| 67 | \( 1 + 8.66T + 67T^{2} \) |
| 71 | \( 1 + 7.54T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 + 4.70T + 79T^{2} \) |
| 83 | \( 1 + 0.268T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 + 19.6T + 97T^{2} \) |
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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
−7.23472029480489271252660103543, −7.08079962857633133153256602400, −6.20398367077282929650537430837, −5.31503124740200586305300822268, −4.79256785490664110623926173749, −3.94210548321619905855715015364, −3.42206897792854004097831340938, −2.96797177600219518048702970611, −1.52450807616281701110377801710, 0,
1.52450807616281701110377801710, 2.96797177600219518048702970611, 3.42206897792854004097831340938, 3.94210548321619905855715015364, 4.79256785490664110623926173749, 5.31503124740200586305300822268, 6.20398367077282929650537430837, 7.08079962857633133153256602400, 7.23472029480489271252660103543