| L(s) = 1 | − 3-s − 3·7-s + 9-s + 11-s − 13-s + 3·17-s + 4·19-s + 3·21-s − 2·23-s − 27-s − 3·29-s + 5·31-s − 33-s − 2·37-s + 39-s − 2·43-s − 9·47-s + 2·49-s − 3·51-s − 5·53-s − 4·57-s + 7·59-s + 11·61-s − 3·63-s − 3·67-s + 2·69-s + 8·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s + 0.727·17-s + 0.917·19-s + 0.654·21-s − 0.417·23-s − 0.192·27-s − 0.557·29-s + 0.898·31-s − 0.174·33-s − 0.328·37-s + 0.160·39-s − 0.304·43-s − 1.31·47-s + 2/7·49-s − 0.420·51-s − 0.686·53-s − 0.529·57-s + 0.911·59-s + 1.40·61-s − 0.377·63-s − 0.366·67-s + 0.240·69-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.55342669219285, −13.88132915509034, −13.56304566158726, −12.80137470684235, −12.59991335453061, −11.97613396443978, −11.54266572039987, −11.10427555468046, −10.25422268633111, −9.926277656375550, −9.605626767358812, −9.041061834416147, −8.198798348440544, −7.819295207138883, −7.039580260959303, −6.659059730137149, −6.196603828081157, −5.462171137180479, −5.165107490460821, −4.329021751166301, −3.640985711603365, −3.227631997549063, −2.487134673457563, −1.585047014697657, −0.8199800799114834, 0,
0.8199800799114834, 1.585047014697657, 2.487134673457563, 3.227631997549063, 3.640985711603365, 4.329021751166301, 5.165107490460821, 5.462171137180479, 6.196603828081157, 6.659059730137149, 7.039580260959303, 7.819295207138883, 8.198798348440544, 9.041061834416147, 9.605626767358812, 9.926277656375550, 10.25422268633111, 11.10427555468046, 11.54266572039987, 11.97613396443978, 12.59991335453061, 12.80137470684235, 13.56304566158726, 13.88132915509034, 14.55342669219285
