L(s) = 1 | − 2.41·2-s + 3.82·4-s + 5-s − 2·7-s − 4.41·8-s − 2.41·10-s − 11-s − 1.17·13-s + 4.82·14-s + 2.99·16-s − 6.82·17-s + 3.82·20-s + 2.41·22-s + 2.82·23-s + 25-s + 2.82·26-s − 7.65·28-s + 3.65·29-s + 1.58·32-s + 16.4·34-s − 2·35-s − 7.65·37-s − 4.41·40-s − 6·41-s − 6·43-s − 3.82·44-s − 6.82·46-s + ⋯ |
L(s) = 1 | − 1.70·2-s + 1.91·4-s + 0.447·5-s − 0.755·7-s − 1.56·8-s − 0.763·10-s − 0.301·11-s − 0.324·13-s + 1.29·14-s + 0.749·16-s − 1.65·17-s + 0.856·20-s + 0.514·22-s + 0.589·23-s + 0.200·25-s + 0.554·26-s − 1.44·28-s + 0.679·29-s + 0.280·32-s + 2.82·34-s − 0.338·35-s − 1.25·37-s − 0.697·40-s − 0.937·41-s − 0.914·43-s − 0.577·44-s − 1.00·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 13 | \( 1 + 1.17T + 13T^{2} \) |
| 17 | \( 1 + 6.82T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + 1.65T + 59T^{2} \) |
| 61 | \( 1 + 9.31T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 1.17T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 - 3.65T + 97T^{2} \) |
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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
−10.32470334397441228459577658957, −9.504751231990373787742277824046, −8.899182679847254668894049577711, −8.042118646857261565031079837643, −6.86104477989725159190689622839, −6.45030383076292275925719248690, −4.88419799925923978199403676724, −3.01478731580852465028396046576, −1.83340295695726343730774282839, 0,
1.83340295695726343730774282839, 3.01478731580852465028396046576, 4.88419799925923978199403676724, 6.45030383076292275925719248690, 6.86104477989725159190689622839, 8.042118646857261565031079837643, 8.899182679847254668894049577711, 9.504751231990373787742277824046, 10.32470334397441228459577658957