| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 11-s + 2·13-s + 15-s − 6·17-s − 4·19-s − 21-s − 8·23-s + 25-s − 27-s − 6·29-s − 8·31-s − 33-s − 35-s − 6·37-s − 2·39-s − 6·41-s − 4·43-s − 45-s + 8·47-s + 49-s + 6·51-s + 10·53-s − 55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s + 0.258·15-s − 1.45·17-s − 0.917·19-s − 0.218·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.174·33-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 + 0.840·51-s + 1.37·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 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 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.69119163265100, −14.04318223623326, −13.44684482767542, −13.17687352446507, −12.46318967200526, −12.02684828291151, −11.57850098560110, −11.11034626198997, −10.54251225992077, −10.40100401197603, −9.386295347797549, −9.039772557181851, −8.412906573634268, −8.067968788734019, −7.193432799941616, −6.952535092708771, −6.268191180884397, −5.746674261296789, −5.232999204072380, −4.478286522081902, −3.876767827866632, −3.791010587619063, −2.569872855048356, −1.892763206927182, −1.426117201744506, 0, 0,
1.426117201744506, 1.892763206927182, 2.569872855048356, 3.791010587619063, 3.876767827866632, 4.478286522081902, 5.232999204072380, 5.746674261296789, 6.268191180884397, 6.952535092708771, 7.193432799941616, 8.067968788734019, 8.412906573634268, 9.039772557181851, 9.386295347797549, 10.40100401197603, 10.54251225992077, 11.11034626198997, 11.57850098560110, 12.02684828291151, 12.46318967200526, 13.17687352446507, 13.44684482767542, 14.04318223623326, 14.69119163265100
