| L(s) = 1 | + 3-s + 5-s + 9-s + 4·11-s − 2·13-s + 15-s − 2·17-s + 8·19-s − 4·23-s + 25-s + 27-s + 6·29-s + 4·33-s − 2·37-s − 2·39-s − 6·41-s + 4·43-s + 45-s + 12·47-s − 7·49-s − 2·51-s + 6·53-s + 4·55-s + 8·57-s + 12·59-s − 14·61-s − 2·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.258·15-s − 0.485·17-s + 1.83·19-s − 0.834·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.696·33-s − 0.328·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s + 1.75·47-s − 49-s − 0.280·51-s + 0.824·53-s + 0.539·55-s + 1.05·57-s + 1.56·59-s − 1.79·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.248403919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.248403919\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−9.806685325851471329154519037796, −9.286569023282875813739958653262, −8.485922182604829094350178884864, −7.46133055491293853299357806271, −6.73746543691429154932020719885, −5.73892987388308473685173648012, −4.66281454287229345249472136354, −3.64257092304627787883654309435, −2.56100826409250417816310835164, −1.30789464322825876552451590970,
1.30789464322825876552451590970, 2.56100826409250417816310835164, 3.64257092304627787883654309435, 4.66281454287229345249472136354, 5.73892987388308473685173648012, 6.73746543691429154932020719885, 7.46133055491293853299357806271, 8.485922182604829094350178884864, 9.286569023282875813739958653262, 9.806685325851471329154519037796
