L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s + 4·11-s + 15-s − 2·17-s + 19-s + 2·21-s − 2·23-s + 25-s − 27-s − 6·29-s − 4·33-s + 2·35-s − 8·37-s − 2·41-s − 6·43-s − 45-s − 2·47-s − 3·49-s + 2·51-s − 4·53-s − 4·55-s − 57-s + 4·59-s − 10·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 1.20·11-s + 0.258·15-s − 0.485·17-s + 0.229·19-s + 0.436·21-s − 0.417·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.696·33-s + 0.338·35-s − 1.31·37-s − 0.312·41-s − 0.914·43-s − 0.149·45-s − 0.291·47-s − 3/7·49-s + 0.280·51-s − 0.549·53-s − 0.539·55-s − 0.132·57-s + 0.520·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−9.417722220576178541041313097075, −8.726288678154087164643012488321, −7.60149507583564019968198836746, −6.74912643139362073039435224031, −6.19840300109211658926543298988, −5.10489514704650899732698905567, −4.06042744379607192509915812992, −3.29476117450426254916209893078, −1.64319018120485410147094923035, 0,
1.64319018120485410147094923035, 3.29476117450426254916209893078, 4.06042744379607192509915812992, 5.10489514704650899732698905567, 6.19840300109211658926543298988, 6.74912643139362073039435224031, 7.60149507583564019968198836746, 8.726288678154087164643012488321, 9.417722220576178541041313097075