| L(s) = 1 | + 3-s + 5-s + 4·7-s + 9-s − 2·13-s + 15-s − 4·19-s + 4·21-s + 25-s + 27-s + 2·29-s + 10·31-s + 4·35-s − 10·37-s − 2·39-s + 6·41-s − 12·43-s + 45-s − 6·47-s + 9·49-s + 12·53-s − 4·57-s + 12·59-s − 8·61-s + 4·63-s − 2·65-s − 8·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s − 0.917·19-s + 0.872·21-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 1.79·31-s + 0.676·35-s − 1.64·37-s − 0.320·39-s + 0.937·41-s − 1.82·43-s + 0.149·45-s − 0.875·47-s + 9/7·49-s + 1.64·53-s − 0.529·57-s + 1.56·59-s − 1.02·61-s + 0.503·63-s − 0.248·65-s − 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.39963512841865, −14.00452974225656, −13.44236563600829, −13.09404886308399, −12.33255870138229, −11.82239168361110, −11.55189109483127, −10.75153291481156, −10.24600168807561, −10.04855443214282, −9.202205408585576, −8.627975689772248, −8.329980893518982, −7.910013305204681, −7.151520311897971, −6.749950969010355, −6.080923041463103, −5.225496931582202, −4.991591975653147, −4.301375118806974, −3.840993025122590, −2.739957432830279, −2.512633036396696, −1.599335668956712, −1.296542578063524, 0,
1.296542578063524, 1.599335668956712, 2.512633036396696, 2.739957432830279, 3.840993025122590, 4.301375118806974, 4.991591975653147, 5.225496931582202, 6.080923041463103, 6.749950969010355, 7.151520311897971, 7.910013305204681, 8.329980893518982, 8.627975689772248, 9.202205408585576, 10.04855443214282, 10.24600168807561, 10.75153291481156, 11.55189109483127, 11.82239168361110, 12.33255870138229, 13.09404886308399, 13.44236563600829, 14.00452974225656, 14.39963512841865
