| L(s) = 1 | + 0.583·2-s − 1.65·4-s − 0.00312·5-s − 2.13·8-s − 0.00182·10-s − 4.94·11-s + 13-s + 2.07·16-s + 5.87·17-s + 5.38·19-s + 0.00519·20-s − 2.88·22-s − 4.23·23-s − 4.99·25-s + 0.583·26-s + 8.87·29-s − 5.36·31-s + 5.48·32-s + 3.43·34-s − 4.67·37-s + 3.14·38-s + 0.00668·40-s − 5.05·41-s + 11.5·43-s + 8.20·44-s − 2.46·46-s + 1.95·47-s + ⋯ |
| L(s) = 1 | + 0.412·2-s − 0.829·4-s − 0.00139·5-s − 0.755·8-s − 0.000577·10-s − 1.49·11-s + 0.277·13-s + 0.518·16-s + 1.42·17-s + 1.23·19-s + 0.00116·20-s − 0.615·22-s − 0.882·23-s − 0.999·25-s + 0.114·26-s + 1.64·29-s − 0.964·31-s + 0.968·32-s + 0.588·34-s − 0.769·37-s + 0.510·38-s + 0.00105·40-s − 0.789·41-s + 1.76·43-s + 1.23·44-s − 0.364·46-s + 0.284·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 0.583T + 2T^{2} \) |
| 5 | \( 1 + 0.00312T + 5T^{2} \) |
| 11 | \( 1 + 4.94T + 11T^{2} \) |
| 17 | \( 1 - 5.87T + 17T^{2} \) |
| 19 | \( 1 - 5.38T + 19T^{2} \) |
| 23 | \( 1 + 4.23T + 23T^{2} \) |
| 29 | \( 1 - 8.87T + 29T^{2} \) |
| 31 | \( 1 + 5.36T + 31T^{2} \) |
| 37 | \( 1 + 4.67T + 37T^{2} \) |
| 41 | \( 1 + 5.05T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 - 1.95T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 - 6.08T + 59T^{2} \) |
| 61 | \( 1 - 9.96T + 61T^{2} \) |
| 67 | \( 1 - 5.00T + 67T^{2} \) |
| 71 | \( 1 + 9.08T + 71T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 + 4.62T + 83T^{2} \) |
| 89 | \( 1 + 5.54T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−7.930714442140383202756211455102, −7.19024436685164387200938695461, −5.98902807391960222904929011635, −5.49495992042771558809749552176, −5.02223075746954418317894083503, −4.04274598727199636725023599784, −3.34036971670311239933780657285, −2.60811834821611335556670557077, −1.21714507743817447388971584550, 0,
1.21714507743817447388971584550, 2.60811834821611335556670557077, 3.34036971670311239933780657285, 4.04274598727199636725023599784, 5.02223075746954418317894083503, 5.49495992042771558809749552176, 5.98902807391960222904929011635, 7.19024436685164387200938695461, 7.930714442140383202756211455102
