| L(s) = 1 | + 2.10·2-s + 0.753·3-s + 2.43·4-s + 5-s + 1.58·6-s + 0.908·8-s − 2.43·9-s + 2.10·10-s − 1.51·11-s + 1.83·12-s − 2.34·13-s + 0.753·15-s − 2.95·16-s + 4.11·17-s − 5.11·18-s + 0.0792·19-s + 2.43·20-s − 3.19·22-s − 4.37·23-s + 0.685·24-s + 25-s − 4.92·26-s − 4.09·27-s − 3.11·29-s + 1.58·30-s + 3.50·31-s − 8.02·32-s + ⋯ |
| L(s) = 1 | + 1.48·2-s + 0.435·3-s + 1.21·4-s + 0.447·5-s + 0.647·6-s + 0.321·8-s − 0.810·9-s + 0.665·10-s − 0.457·11-s + 0.529·12-s − 0.649·13-s + 0.194·15-s − 0.737·16-s + 0.999·17-s − 1.20·18-s + 0.0181·19-s + 0.543·20-s − 0.681·22-s − 0.911·23-s + 0.139·24-s + 0.200·25-s − 0.966·26-s − 0.787·27-s − 0.579·29-s + 0.289·30-s + 0.629·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 - 2.10T + 2T^{2} \) |
| 3 | \( 1 - 0.753T + 3T^{2} \) |
| 11 | \( 1 + 1.51T + 11T^{2} \) |
| 13 | \( 1 + 2.34T + 13T^{2} \) |
| 17 | \( 1 - 4.11T + 17T^{2} \) |
| 19 | \( 1 - 0.0792T + 19T^{2} \) |
| 23 | \( 1 + 4.37T + 23T^{2} \) |
| 29 | \( 1 + 3.11T + 29T^{2} \) |
| 31 | \( 1 - 3.50T + 31T^{2} \) |
| 41 | \( 1 - 0.182T + 41T^{2} \) |
| 43 | \( 1 + 9.09T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 - 0.660T + 53T^{2} \) |
| 59 | \( 1 + 8.34T + 59T^{2} \) |
| 61 | \( 1 - 3.85T + 61T^{2} \) |
| 67 | \( 1 + 3.85T + 67T^{2} \) |
| 71 | \( 1 + 0.0167T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 - 3.06T + 79T^{2} \) |
| 83 | \( 1 + 5.24T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−7.23054226448434850919525546025, −6.42611435297302649970294539701, −5.82121536077672233966322652143, −5.26803312831383002357785917921, −4.72762895738577502601355823126, −3.75176357540663585315593261295, −3.15096658968970221123927481434, −2.54642831724651551139609394096, −1.74549764661476491648731500442, 0,
1.74549764661476491648731500442, 2.54642831724651551139609394096, 3.15096658968970221123927481434, 3.75176357540663585315593261295, 4.72762895738577502601355823126, 5.26803312831383002357785917921, 5.82121536077672233966322652143, 6.42611435297302649970294539701, 7.23054226448434850919525546025
