L(s) = 1 | + 3-s + 5-s + 4·7-s + 9-s − 4·11-s + 2·13-s + 15-s − 2·17-s + 4·21-s + 8·23-s + 25-s + 27-s + 29-s − 4·33-s + 4·35-s − 6·37-s + 2·39-s − 2·41-s − 4·43-s + 45-s + 8·47-s + 9·49-s − 2·51-s + 14·53-s − 4·55-s − 12·59-s + 10·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.258·15-s − 0.485·17-s + 0.872·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s + 0.185·29-s − 0.696·33-s + 0.676·35-s − 0.986·37-s + 0.320·39-s − 0.312·41-s − 0.609·43-s + 0.149·45-s + 1.16·47-s + 9/7·49-s − 0.280·51-s + 1.92·53-s − 0.539·55-s − 1.56·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.453849916\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.453849916\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 29 | \( 1 - T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−8.041690072857629571250212636313, −7.34649545966690325753992727808, −6.74535728402544767266415218142, −5.60550991202005754496990413196, −5.10911532275526578852361659096, −4.50967520620296409526271031901, −3.49636907865512559652923527455, −2.57085677814445908706293080760, −1.92648890475029961552591597282, −0.965715644511385356285103984125,
0.965715644511385356285103984125, 1.92648890475029961552591597282, 2.57085677814445908706293080760, 3.49636907865512559652923527455, 4.50967520620296409526271031901, 5.10911532275526578852361659096, 5.60550991202005754496990413196, 6.74535728402544767266415218142, 7.34649545966690325753992727808, 8.041690072857629571250212636313