| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s − 4·11-s + 2·13-s − 15-s + 2·17-s − 4·19-s + 21-s − 23-s + 25-s + 27-s + 6·29-s + 8·31-s − 4·33-s − 35-s − 6·37-s + 2·39-s + 6·41-s − 4·43-s − 45-s + 8·47-s + 49-s + 2·51-s − 6·53-s + 4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.258·15-s + 0.485·17-s − 0.917·19-s + 0.218·21-s − 0.208·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.696·33-s − 0.169·35-s − 0.986·37-s + 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s + 1/7·49-s + 0.280·51-s − 0.824·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−13.66746324977687, −12.99681176319944, −12.68660183515094, −12.15759808370566, −11.73575179866221, −11.05347453261237, −10.64406845557360, −10.26642887091767, −9.827998264542892, −9.078974624754624, −8.581704314623767, −8.207140903968354, −7.904548272032935, −7.329423781654147, −6.789962198732828, −6.144773928921121, −5.698216117665121, −4.923517386393996, −4.541595461501124, −4.057572846717482, −3.290621319950710, −2.855847680945316, −2.322339088617112, −1.569874362699252, −0.8765062734796305, 0,
0.8765062734796305, 1.569874362699252, 2.322339088617112, 2.855847680945316, 3.290621319950710, 4.057572846717482, 4.541595461501124, 4.923517386393996, 5.698216117665121, 6.144773928921121, 6.789962198732828, 7.329423781654147, 7.904548272032935, 8.207140903968354, 8.581704314623767, 9.078974624754624, 9.827998264542892, 10.26642887091767, 10.64406845557360, 11.05347453261237, 11.73575179866221, 12.15759808370566, 12.68660183515094, 12.99681176319944, 13.66746324977687
