L(s) = 1 | − 2-s + 4-s + 5-s + 4·7-s − 8-s − 10-s − 6·11-s − 4·14-s + 16-s + 6·17-s − 2·19-s + 20-s + 6·22-s − 6·23-s + 25-s + 4·28-s + 6·29-s − 2·31-s − 32-s − 6·34-s + 4·35-s − 2·37-s + 2·38-s − 40-s − 6·41-s + 2·43-s − 6·44-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.51·7-s − 0.353·8-s − 0.316·10-s − 1.80·11-s − 1.06·14-s + 1/4·16-s + 1.45·17-s − 0.458·19-s + 0.223·20-s + 1.27·22-s − 1.25·23-s + 1/5·25-s + 0.755·28-s + 1.11·29-s − 0.359·31-s − 0.176·32-s − 1.02·34-s + 0.676·35-s − 0.328·37-s + 0.324·38-s − 0.158·40-s − 0.937·41-s + 0.304·43-s − 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−16.22668538402631, −15.98311632585823, −15.16904467896952, −14.72018969332716, −14.12910210740831, −13.68474520927649, −12.86824562805226, −12.32286296081836, −11.74339137911533, −11.07155521526933, −10.59683043624435, −10.02527603880378, −9.738369955409849, −8.585952299050329, −8.146911095995778, −7.953792045996156, −7.238134956913372, −6.390487263226516, −5.543820401786759, −5.212029930095252, −4.532420674125924, −3.434538690442281, −2.589993187455778, −1.932266307946935, −1.229665691730756, 0,
1.229665691730756, 1.932266307946935, 2.589993187455778, 3.434538690442281, 4.532420674125924, 5.212029930095252, 5.543820401786759, 6.390487263226516, 7.238134956913372, 7.953792045996156, 8.146911095995778, 8.585952299050329, 9.738369955409849, 10.02527603880378, 10.59683043624435, 11.07155521526933, 11.74339137911533, 12.32286296081836, 12.86824562805226, 13.68474520927649, 14.12910210740831, 14.72018969332716, 15.16904467896952, 15.98311632585823, 16.22668538402631