| L(s) = 1 | − 2-s − 4-s + 2·7-s + 3·8-s + 2·11-s − 2·13-s − 2·14-s − 16-s − 2·17-s + 19-s − 2·22-s + 2·26-s − 2·28-s − 6·29-s − 4·31-s − 5·32-s + 2·34-s + 2·37-s − 38-s + 2·41-s − 10·43-s − 2·44-s − 3·49-s + 2·52-s + 10·53-s + 6·56-s + 6·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.755·7-s + 1.06·8-s + 0.603·11-s − 0.554·13-s − 0.534·14-s − 1/4·16-s − 0.485·17-s + 0.229·19-s − 0.426·22-s + 0.392·26-s − 0.377·28-s − 1.11·29-s − 0.718·31-s − 0.883·32-s + 0.342·34-s + 0.328·37-s − 0.162·38-s + 0.312·41-s − 1.52·43-s − 0.301·44-s − 3/7·49-s + 0.277·52-s + 1.37·53-s + 0.801·56-s + 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.050247046324566422453399051985, −7.51395825732491357428185619721, −6.81038038339561258516758833543, −5.73220954205655863304800274764, −4.95232142393563803847229049406, −4.33220688169430552857891111530, −3.47622786805106583216168399871, −2.11075451998304823140925559413, −1.31269382709472943429853389134, 0,
1.31269382709472943429853389134, 2.11075451998304823140925559413, 3.47622786805106583216168399871, 4.33220688169430552857891111530, 4.95232142393563803847229049406, 5.73220954205655863304800274764, 6.81038038339561258516758833543, 7.51395825732491357428185619721, 8.050247046324566422453399051985
