| L(s) = 1 | − 2.34·2-s + 3.48·4-s − 1.34·5-s − 3.48·8-s + 3.14·10-s − 1.14·11-s − 13-s + 1.19·16-s + 5.83·17-s + 3.34·19-s − 4.68·20-s + 2.68·22-s + 3.17·23-s − 3.19·25-s + 2.34·26-s − 10.4·29-s − 1.63·31-s + 4.17·32-s − 13.6·34-s + 8.51·37-s − 7.83·38-s + 4.68·40-s − 0.292·41-s − 8.15·43-s − 4.00·44-s − 7.43·46-s − 10.6·47-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + 1.74·4-s − 0.600·5-s − 1.23·8-s + 0.994·10-s − 0.345·11-s − 0.277·13-s + 0.299·16-s + 1.41·17-s + 0.766·19-s − 1.04·20-s + 0.572·22-s + 0.662·23-s − 0.639·25-s + 0.459·26-s − 1.94·29-s − 0.293·31-s + 0.738·32-s − 2.34·34-s + 1.40·37-s − 1.27·38-s + 0.740·40-s − 0.0457·41-s − 1.24·43-s − 0.603·44-s − 1.09·46-s − 1.54·47-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 | \( 1 + 2.34T + 2T^{2} \) |
| 5 | \( 1 + 1.34T + 5T^{2} \) |
| 11 | \( 1 + 1.14T + 11T^{2} \) |
| 17 | \( 1 - 5.83T + 17T^{2} \) |
| 19 | \( 1 - 3.34T + 19T^{2} \) |
| 23 | \( 1 - 3.17T + 23T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 + 1.63T + 31T^{2} \) |
| 37 | \( 1 - 8.51T + 37T^{2} \) |
| 41 | \( 1 + 0.292T + 41T^{2} \) |
| 43 | \( 1 + 8.15T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 - 0.782T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 6.10T + 67T^{2} \) |
| 71 | \( 1 + 1.53T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 - 0.882T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 5.73T + 89T^{2} \) |
| 97 | \( 1 - 5.34T + 97T^{2} \) |
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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
−7.82430375515476795191906851658, −7.45301200944173434401777577436, −6.74806846502437154940726323465, −5.72927753631409768565412057026, −5.03536941510938430234784295977, −3.81766842303496704797449335203, −3.06182567836368138698080706012, −2.00141128250161081989778808507, −1.05568239992044774063077333034, 0,
1.05568239992044774063077333034, 2.00141128250161081989778808507, 3.06182567836368138698080706012, 3.81766842303496704797449335203, 5.03536941510938430234784295977, 5.72927753631409768565412057026, 6.74806846502437154940726323465, 7.45301200944173434401777577436, 7.82430375515476795191906851658
