| L(s) = 1 | + 3·3-s − 3·5-s + 7-s + 6·9-s − 4·11-s − 9·15-s + 5·17-s − 6·19-s + 3·21-s − 6·23-s + 4·25-s + 9·27-s − 4·29-s − 12·33-s − 3·35-s − 3·37-s + 12·41-s − 3·43-s − 18·45-s − 7·47-s − 6·49-s + 15·51-s − 2·53-s + 12·55-s − 18·57-s + 2·59-s − 12·61-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.34·5-s + 0.377·7-s + 2·9-s − 1.20·11-s − 2.32·15-s + 1.21·17-s − 1.37·19-s + 0.654·21-s − 1.25·23-s + 4/5·25-s + 1.73·27-s − 0.742·29-s − 2.08·33-s − 0.507·35-s − 0.493·37-s + 1.87·41-s − 0.457·43-s − 2.68·45-s − 1.02·47-s − 6/7·49-s + 2.10·51-s − 0.274·53-s + 1.61·55-s − 2.38·57-s + 0.260·59-s − 1.53·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 11 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−7.925951474418443176738486971457, −7.68198838538682480497091541735, −6.67881212838622481253285158381, −5.53166881680342542228902308834, −4.50845484908740139404107116093, −3.95879686429694165893222914306, −3.28471184639679344045513309661, −2.52948032722165475441830069214, −1.64654038590767636903333916792, 0,
1.64654038590767636903333916792, 2.52948032722165475441830069214, 3.28471184639679344045513309661, 3.95879686429694165893222914306, 4.50845484908740139404107116093, 5.53166881680342542228902308834, 6.67881212838622481253285158381, 7.68198838538682480497091541735, 7.925951474418443176738486971457
