| L(s) = 1 | + 3.96·5-s − 0.561·7-s + 0.868·11-s − 4.12·13-s − 3.09·17-s + 4.56·19-s + 9.27·23-s + 10.6·25-s + 2.22·29-s + 5.56·31-s − 2.22·35-s + 9.12·37-s − 4.82·41-s − 43-s − 3.96·47-s − 6.68·49-s − 11.0·53-s + 3.43·55-s + 1.73·59-s − 5.12·61-s − 16.3·65-s + 12.6·67-s − 1.73·71-s + 9.12·73-s − 0.487·77-s + 2.87·79-s + 11.5·83-s + ⋯ |
| L(s) = 1 | + 1.77·5-s − 0.212·7-s + 0.261·11-s − 1.14·13-s − 0.749·17-s + 1.04·19-s + 1.93·23-s + 2.13·25-s + 0.412·29-s + 0.998·31-s − 0.375·35-s + 1.49·37-s − 0.754·41-s − 0.152·43-s − 0.577·47-s − 0.954·49-s − 1.51·53-s + 0.463·55-s + 0.226·59-s − 0.655·61-s − 2.02·65-s + 1.54·67-s − 0.206·71-s + 1.06·73-s − 0.0555·77-s + 0.323·79-s + 1.26·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.352432068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.352432068\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 5 | \( 1 - 3.96T + 5T^{2} \) |
| 7 | \( 1 + 0.561T + 7T^{2} \) |
| 11 | \( 1 - 0.868T + 11T^{2} \) |
| 13 | \( 1 + 4.12T + 13T^{2} \) |
| 17 | \( 1 + 3.09T + 17T^{2} \) |
| 19 | \( 1 - 4.56T + 19T^{2} \) |
| 23 | \( 1 - 9.27T + 23T^{2} \) |
| 29 | \( 1 - 2.22T + 29T^{2} \) |
| 31 | \( 1 - 5.56T + 31T^{2} \) |
| 37 | \( 1 - 9.12T + 37T^{2} \) |
| 41 | \( 1 + 4.82T + 41T^{2} \) |
| 47 | \( 1 + 3.96T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 - 1.73T + 59T^{2} \) |
| 61 | \( 1 + 5.12T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 + 1.73T + 71T^{2} \) |
| 73 | \( 1 - 9.12T + 73T^{2} \) |
| 79 | \( 1 - 2.87T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 + 9.65T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−9.546040053565666948853937819053, −8.975726984499767125205740528260, −7.82165811157363302908096804163, −6.71390122499470485858431099332, −6.38240959857794079661842159838, −5.17015622311886895039432289624, −4.82039401544167605299535122525, −3.09597826360840128614218757192, −2.37336296519078491719381575674, −1.17665072505418913914486507575,
1.17665072505418913914486507575, 2.37336296519078491719381575674, 3.09597826360840128614218757192, 4.82039401544167605299535122525, 5.17015622311886895039432289624, 6.38240959857794079661842159838, 6.71390122499470485858431099332, 7.82165811157363302908096804163, 8.975726984499767125205740528260, 9.546040053565666948853937819053
