| L(s) = 1 | − 5-s − 3.12·7-s + 4·11-s + 3.56·13-s − 5.12·17-s + 4·19-s − 23-s + 25-s + 4.43·29-s + 5.56·31-s + 3.12·35-s + 1.12·37-s + 3.56·41-s − 0.876·43-s − 8.68·47-s + 2.75·49-s − 12.2·53-s − 4·55-s − 10.2·59-s + 2.87·61-s − 3.56·65-s − 10.2·67-s + 8.68·71-s + 12.4·73-s − 12.4·77-s + 6.24·79-s − 12·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.18·7-s + 1.20·11-s + 0.987·13-s − 1.24·17-s + 0.917·19-s − 0.208·23-s + 0.200·25-s + 0.824·29-s + 0.998·31-s + 0.527·35-s + 0.184·37-s + 0.556·41-s − 0.133·43-s − 1.26·47-s + 0.393·49-s − 1.68·53-s − 0.539·55-s − 1.33·59-s + 0.368·61-s − 0.441·65-s − 1.25·67-s + 1.03·71-s + 1.45·73-s − 1.42·77-s + 0.702·79-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.600087987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.600087987\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 3.12T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 3.56T + 13T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 29 | \( 1 - 4.43T + 29T^{2} \) |
| 31 | \( 1 - 5.56T + 31T^{2} \) |
| 37 | \( 1 - 1.12T + 37T^{2} \) |
| 41 | \( 1 - 3.56T + 41T^{2} \) |
| 43 | \( 1 + 0.876T + 43T^{2} \) |
| 47 | \( 1 + 8.68T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 2.87T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 8.68T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 - 6.24T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 0.246T + 97T^{2} \) |
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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
−7.84707399761984396157813710519, −6.90898529115688709566366017144, −6.39337597931529919222221143614, −6.10199287219393165201701679281, −4.84551047265947340719162026368, −4.20855897884656671245321242527, −3.43834793725010218979562495934, −2.92906737971132218908179495127, −1.64704929413425993711186478908, −0.63811701482272834039898789466,
0.63811701482272834039898789466, 1.64704929413425993711186478908, 2.92906737971132218908179495127, 3.43834793725010218979562495934, 4.20855897884656671245321242527, 4.84551047265947340719162026368, 6.10199287219393165201701679281, 6.39337597931529919222221143614, 6.90898529115688709566366017144, 7.84707399761984396157813710519
