| L(s) = 1 | − 3·3-s − 3·5-s + 17·7-s + 9·9-s + 19·11-s − 30·13-s + 9·15-s − 97·17-s − 19·19-s − 51·21-s + 28·23-s − 116·25-s − 27·27-s + 126·29-s + 126·31-s − 57·33-s − 51·35-s + 64·37-s + 90·39-s + 80·41-s + 453·43-s − 27·45-s − 107·47-s − 54·49-s + 291·51-s − 326·53-s − 57·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.268·5-s + 0.917·7-s + 1/3·9-s + 0.520·11-s − 0.640·13-s + 0.154·15-s − 1.38·17-s − 0.229·19-s − 0.529·21-s + 0.253·23-s − 0.927·25-s − 0.192·27-s + 0.806·29-s + 0.730·31-s − 0.300·33-s − 0.246·35-s + 0.284·37-s + 0.369·39-s + 0.304·41-s + 1.60·43-s − 0.0894·45-s − 0.332·47-s − 0.157·49-s + 0.798·51-s − 0.844·53-s − 0.139·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 19 | \( 1 + p T \) |
| good | 5 | \( 1 + 3 T + p^{3} T^{2} \) |
| 7 | \( 1 - 17 T + p^{3} T^{2} \) |
| 11 | \( 1 - 19 T + p^{3} T^{2} \) |
| 13 | \( 1 + 30 T + p^{3} T^{2} \) |
| 17 | \( 1 + 97 T + p^{3} T^{2} \) |
| 23 | \( 1 - 28 T + p^{3} T^{2} \) |
| 29 | \( 1 - 126 T + p^{3} T^{2} \) |
| 31 | \( 1 - 126 T + p^{3} T^{2} \) |
| 37 | \( 1 - 64 T + p^{3} T^{2} \) |
| 41 | \( 1 - 80 T + p^{3} T^{2} \) |
| 43 | \( 1 - 453 T + p^{3} T^{2} \) |
| 47 | \( 1 + 107 T + p^{3} T^{2} \) |
| 53 | \( 1 + 326 T + p^{3} T^{2} \) |
| 59 | \( 1 + 56 T + p^{3} T^{2} \) |
| 61 | \( 1 - 47 T + p^{3} T^{2} \) |
| 67 | \( 1 - 168 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1060 T + p^{3} T^{2} \) |
| 73 | \( 1 + 659 T + p^{3} T^{2} \) |
| 79 | \( 1 + 592 T + p^{3} T^{2} \) |
| 83 | \( 1 + 892 T + p^{3} T^{2} \) |
| 89 | \( 1 + 310 T + p^{3} T^{2} \) |
| 97 | \( 1 + 874 T + p^{3} T^{2} \) |
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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
−9.285986064513183790119400990515, −8.415556474451176125783821769913, −7.55981883128693469580535508875, −6.70311849596944819719127245281, −5.81820827226268019823869428806, −4.66157544229334042133303095629, −4.23034062334052633511294845760, −2.59069155603486382634601969287, −1.39324928735504173135287126945, 0,
1.39324928735504173135287126945, 2.59069155603486382634601969287, 4.23034062334052633511294845760, 4.66157544229334042133303095629, 5.81820827226268019823869428806, 6.70311849596944819719127245281, 7.55981883128693469580535508875, 8.415556474451176125783821769913, 9.285986064513183790119400990515
