| L(s) = 1 | + 3-s − 5-s + 3·7-s + 9-s − 2.31·11-s + 4.31·13-s − 15-s + 3.31·17-s + 8.31·19-s + 3·21-s − 23-s + 25-s + 27-s − 7.31·29-s − 3.63·31-s − 2.31·33-s − 3·35-s − 1.63·37-s + 4.31·39-s − 3.31·41-s + 10.6·43-s − 45-s + 8.94·47-s + 2·49-s + 3.31·51-s + 9.94·53-s + 2.31·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.13·7-s + 0.333·9-s − 0.698·11-s + 1.19·13-s − 0.258·15-s + 0.804·17-s + 1.90·19-s + 0.654·21-s − 0.208·23-s + 0.200·25-s + 0.192·27-s − 1.35·29-s − 0.652·31-s − 0.403·33-s − 0.507·35-s − 0.268·37-s + 0.691·39-s − 0.517·41-s + 1.62·43-s − 0.149·45-s + 1.30·47-s + 0.285·49-s + 0.464·51-s + 1.36·53-s + 0.312·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.958510084\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.958510084\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 + 2.31T + 11T^{2} \) |
| 13 | \( 1 - 4.31T + 13T^{2} \) |
| 17 | \( 1 - 3.31T + 17T^{2} \) |
| 19 | \( 1 - 8.31T + 19T^{2} \) |
| 29 | \( 1 + 7.31T + 29T^{2} \) |
| 31 | \( 1 + 3.63T + 31T^{2} \) |
| 37 | \( 1 + 1.63T + 37T^{2} \) |
| 41 | \( 1 + 3.31T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 - 8.94T + 47T^{2} \) |
| 53 | \( 1 - 9.94T + 53T^{2} \) |
| 59 | \( 1 + 7.94T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 15.6T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 - 8.31T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 1.31T + 83T^{2} \) |
| 89 | \( 1 + 4.63T + 89T^{2} \) |
| 97 | \( 1 - 12T + 97T^{2} \) |
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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
−7.895912560611706931511162404638, −7.71656112312670520135656812949, −7.06995594133922418958359461992, −5.62669026024988118858364244114, −5.47889022996322192615554149251, −4.35095757406865914646393323586, −3.65516797609184198728460620158, −2.93281823502459694205191738013, −1.80109921575139812342620603584, −0.965021448672469713572827527866,
0.965021448672469713572827527866, 1.80109921575139812342620603584, 2.93281823502459694205191738013, 3.65516797609184198728460620158, 4.35095757406865914646393323586, 5.47889022996322192615554149251, 5.62669026024988118858364244114, 7.06995594133922418958359461992, 7.71656112312670520135656812949, 7.895912560611706931511162404638
