L(s) = 1 | + 2.04·2-s + 2.19·4-s + 5-s + 0.589·7-s + 0.406·8-s + 2.04·10-s − 1.50·13-s + 1.20·14-s − 3.56·16-s − 4.91·17-s − 6.82·19-s + 2.19·20-s − 0.0822·23-s + 25-s − 3.08·26-s + 1.29·28-s − 8.42·29-s − 1.97·31-s − 8.11·32-s − 10.0·34-s + 0.589·35-s − 3.57·37-s − 13.9·38-s + 0.406·40-s − 4.07·41-s − 4.86·43-s − 0.168·46-s + ⋯ |
L(s) = 1 | + 1.44·2-s + 1.09·4-s + 0.447·5-s + 0.222·7-s + 0.143·8-s + 0.647·10-s − 0.417·13-s + 0.323·14-s − 0.890·16-s − 1.19·17-s − 1.56·19-s + 0.491·20-s − 0.0171·23-s + 0.200·25-s − 0.604·26-s + 0.245·28-s − 1.56·29-s − 0.355·31-s − 1.43·32-s − 1.72·34-s + 0.0997·35-s − 0.588·37-s − 2.26·38-s + 0.0643·40-s − 0.636·41-s − 0.741·43-s − 0.0248·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 2 | \( 1 - 2.04T + 2T^{2} \) |
| 7 | \( 1 - 0.589T + 7T^{2} \) |
| 13 | \( 1 + 1.50T + 13T^{2} \) |
| 17 | \( 1 + 4.91T + 17T^{2} \) |
| 19 | \( 1 + 6.82T + 19T^{2} \) |
| 23 | \( 1 + 0.0822T + 23T^{2} \) |
| 29 | \( 1 + 8.42T + 29T^{2} \) |
| 31 | \( 1 + 1.97T + 31T^{2} \) |
| 37 | \( 1 + 3.57T + 37T^{2} \) |
| 41 | \( 1 + 4.07T + 41T^{2} \) |
| 43 | \( 1 + 4.86T + 43T^{2} \) |
| 47 | \( 1 - 5.77T + 47T^{2} \) |
| 53 | \( 1 - 6.77T + 53T^{2} \) |
| 59 | \( 1 + 5.58T + 59T^{2} \) |
| 61 | \( 1 - 0.960T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 + 1.28T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 4.01T + 83T^{2} \) |
| 89 | \( 1 - 6.08T + 89T^{2} \) |
| 97 | \( 1 + 6.70T + 97T^{2} \) |
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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
−7.59405397072549758261187912913, −6.63388161632371399162630565789, −6.40919393936963363725013578538, −5.38348840328997605507432744572, −4.96708496388070045793540863747, −4.12145306790905091081641213558, −3.55132956922488185625661811382, −2.35709281585516250231458644982, −1.96408019759743233090842600511, 0,
1.96408019759743233090842600511, 2.35709281585516250231458644982, 3.55132956922488185625661811382, 4.12145306790905091081641213558, 4.96708496388070045793540863747, 5.38348840328997605507432744572, 6.40919393936963363725013578538, 6.63388161632371399162630565789, 7.59405397072549758261187912913