| L(s) = 1 | − 1.34·2-s − 0.184·4-s − 2.53·5-s + 7-s + 2.94·8-s + 3.41·10-s − 0.467·11-s + 5.82·13-s − 1.34·14-s − 3.59·16-s − 3.87·17-s − 2.18·19-s + 0.467·20-s + 0.630·22-s + 0.106·23-s + 1.41·25-s − 7.84·26-s − 0.184·28-s − 8.78·29-s − 7.68·31-s − 1.04·32-s + 5.22·34-s − 2.53·35-s − 7.68·37-s + 2.94·38-s − 7.45·40-s + 2.22·41-s + ⋯ |
| L(s) = 1 | − 0.952·2-s − 0.0923·4-s − 1.13·5-s + 0.377·7-s + 1.04·8-s + 1.07·10-s − 0.141·11-s + 1.61·13-s − 0.360·14-s − 0.899·16-s − 0.940·17-s − 0.501·19-s + 0.104·20-s + 0.134·22-s + 0.0221·23-s + 0.282·25-s − 1.53·26-s − 0.0349·28-s − 1.63·29-s − 1.37·31-s − 0.184·32-s + 0.896·34-s − 0.428·35-s − 1.26·37-s + 0.477·38-s − 1.17·40-s + 0.347·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 + 1.34T + 2T^{2} \) |
| 5 | \( 1 + 2.53T + 5T^{2} \) |
| 11 | \( 1 + 0.467T + 11T^{2} \) |
| 13 | \( 1 - 5.82T + 13T^{2} \) |
| 17 | \( 1 + 3.87T + 17T^{2} \) |
| 19 | \( 1 + 2.18T + 19T^{2} \) |
| 23 | \( 1 - 0.106T + 23T^{2} \) |
| 29 | \( 1 + 8.78T + 29T^{2} \) |
| 31 | \( 1 + 7.68T + 31T^{2} \) |
| 37 | \( 1 + 7.68T + 37T^{2} \) |
| 41 | \( 1 - 2.22T + 41T^{2} \) |
| 43 | \( 1 - 1.22T + 43T^{2} \) |
| 47 | \( 1 - 5.33T + 47T^{2} \) |
| 53 | \( 1 - 0.716T + 53T^{2} \) |
| 59 | \( 1 + 0.736T + 59T^{2} \) |
| 61 | \( 1 - 0.958T + 61T^{2} \) |
| 67 | \( 1 + 9.63T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 - 2.73T + 83T^{2} \) |
| 89 | \( 1 - 8.11T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−10.46909880824011790021626696102, −8.983654195024787525592099519702, −8.760614867739603285440135304412, −7.78014400988830651791169088806, −7.13735710448351875663360313624, −5.74036438185373714314048704932, −4.37809598544036147863262519473, −3.69179883501881382005233149482, −1.67718686743966778762820678462, 0,
1.67718686743966778762820678462, 3.69179883501881382005233149482, 4.37809598544036147863262519473, 5.74036438185373714314048704932, 7.13735710448351875663360313624, 7.78014400988830651791169088806, 8.760614867739603285440135304412, 8.983654195024787525592099519702, 10.46909880824011790021626696102
