| L(s) = 1 | − 1.11·3-s − 1.47·5-s + 7-s − 1.75·9-s + 2·11-s + 0.715·13-s + 1.64·15-s − 17-s + 1.28·19-s − 1.11·21-s − 2.22·23-s − 2.83·25-s + 5.30·27-s − 9.17·29-s + 6.75·31-s − 2.22·33-s − 1.47·35-s + 8.45·37-s − 0.798·39-s − 6.77·41-s + 4.39·43-s + 2.58·45-s − 2.22·47-s + 49-s + 1.11·51-s + 6.64·53-s − 2.94·55-s + ⋯ |
| L(s) = 1 | − 0.643·3-s − 0.658·5-s + 0.377·7-s − 0.585·9-s + 0.603·11-s + 0.198·13-s + 0.423·15-s − 0.242·17-s + 0.294·19-s − 0.243·21-s − 0.464·23-s − 0.566·25-s + 1.02·27-s − 1.70·29-s + 1.21·31-s − 0.388·33-s − 0.248·35-s + 1.39·37-s − 0.127·39-s − 1.05·41-s + 0.670·43-s + 0.385·45-s − 0.325·47-s + 0.142·49-s + 0.156·51-s + 0.913·53-s − 0.397·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.060371358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.060371358\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 1.11T + 3T^{2} \) |
| 5 | \( 1 + 1.47T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 0.715T + 13T^{2} \) |
| 19 | \( 1 - 1.28T + 19T^{2} \) |
| 23 | \( 1 + 2.22T + 23T^{2} \) |
| 29 | \( 1 + 9.17T + 29T^{2} \) |
| 31 | \( 1 - 6.75T + 31T^{2} \) |
| 37 | \( 1 - 8.45T + 37T^{2} \) |
| 41 | \( 1 + 6.77T + 41T^{2} \) |
| 43 | \( 1 - 4.39T + 43T^{2} \) |
| 47 | \( 1 + 2.22T + 47T^{2} \) |
| 53 | \( 1 - 6.64T + 53T^{2} \) |
| 59 | \( 1 - 5.28T + 59T^{2} \) |
| 61 | \( 1 - 0.885T + 61T^{2} \) |
| 67 | \( 1 - 4.90T + 67T^{2} \) |
| 71 | \( 1 - 16.4T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 - 2.37T + 83T^{2} \) |
| 89 | \( 1 - 7.51T + 89T^{2} \) |
| 97 | \( 1 - 5.85T + 97T^{2} \) |
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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
−9.185423290276763426995205410469, −8.317024687015370650030560933601, −7.74233846905720909024290165920, −6.75820836361030537740605149948, −5.99806429703410050219075956450, −5.23723913875066293173055510667, −4.26875451237558149414835200495, −3.49991964943572465203415018067, −2.18067002695916024099641855585, −0.71018371877375355214681175087,
0.71018371877375355214681175087, 2.18067002695916024099641855585, 3.49991964943572465203415018067, 4.26875451237558149414835200495, 5.23723913875066293173055510667, 5.99806429703410050219075956450, 6.75820836361030537740605149948, 7.74233846905720909024290165920, 8.317024687015370650030560933601, 9.185423290276763426995205410469
