| L(s) = 1 | + 5-s − 4·11-s − 6·13-s + 6·17-s + 4·19-s + 25-s − 2·29-s − 8·31-s + 2·37-s + 6·41-s − 12·43-s − 8·47-s − 7·49-s + 6·53-s − 4·55-s + 12·59-s − 14·61-s − 6·65-s − 4·67-s − 8·71-s − 6·73-s − 8·79-s − 12·83-s + 6·85-s − 10·89-s + 4·95-s + 2·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.20·11-s − 1.66·13-s + 1.45·17-s + 0.917·19-s + 1/5·25-s − 0.371·29-s − 1.43·31-s + 0.328·37-s + 0.937·41-s − 1.82·43-s − 1.16·47-s − 49-s + 0.824·53-s − 0.539·55-s + 1.56·59-s − 1.79·61-s − 0.744·65-s − 0.488·67-s − 0.949·71-s − 0.702·73-s − 0.900·79-s − 1.31·83-s + 0.650·85-s − 1.05·89-s + 0.410·95-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.281438123975201335840704816115, −7.49319629717735131747504234927, −7.19464749200877042506852879968, −5.86215101035639287486075524635, −5.32415764508171872286866957381, −4.72041309651433423591886250489, −3.35030464779548994692224119619, −2.67630125421591650399057166027, −1.59198724080807310872956999404, 0,
1.59198724080807310872956999404, 2.67630125421591650399057166027, 3.35030464779548994692224119619, 4.72041309651433423591886250489, 5.32415764508171872286866957381, 5.86215101035639287486075524635, 7.19464749200877042506852879968, 7.49319629717735131747504234927, 8.281438123975201335840704816115
