| L(s) = 1 | − 3-s + 2·5-s + 2·7-s + 9-s + 11-s + 4·13-s − 2·15-s + 17-s − 2·19-s − 2·21-s − 2·23-s − 25-s − 27-s + 2·29-s + 4·31-s − 33-s + 4·35-s + 6·37-s − 4·39-s + 6·41-s − 2·43-s + 2·45-s − 3·49-s − 51-s − 12·53-s + 2·55-s + 2·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 0.516·15-s + 0.242·17-s − 0.458·19-s − 0.436·21-s − 0.417·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.718·31-s − 0.174·33-s + 0.676·35-s + 0.986·37-s − 0.640·39-s + 0.937·41-s − 0.304·43-s + 0.298·45-s − 3/7·49-s − 0.140·51-s − 1.64·53-s + 0.269·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.599565247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.599565247\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.88314991589701052915119048979, −6.84280785445301606050522280569, −6.20725559024613781322835018331, −5.84355774232248501121019046253, −5.02605456074928841662469864504, −4.35440870135993346255354999468, −3.57865820150810852691096362857, −2.42296579603665248458932251127, −1.64518280222280485175771429383, −0.862043711233310762961729495085,
0.862043711233310762961729495085, 1.64518280222280485175771429383, 2.42296579603665248458932251127, 3.57865820150810852691096362857, 4.35440870135993346255354999468, 5.02605456074928841662469864504, 5.84355774232248501121019046253, 6.20725559024613781322835018331, 6.84280785445301606050522280569, 7.88314991589701052915119048979
