| L(s) = 1 | + 0.325·3-s − 2.32·5-s + 3.24·7-s − 2.89·9-s + 3.01·11-s − 3.93·13-s − 0.757·15-s + 2·17-s + 0.325·19-s + 1.05·21-s + 1.63·23-s + 0.409·25-s − 1.92·27-s − 7.70·29-s + 1.81·31-s + 0.983·33-s − 7.54·35-s − 9.01·37-s − 1.28·39-s + 1.24·41-s + 9.68·43-s + 6.73·45-s + 0.179·47-s + 3.51·49-s + 0.651·51-s − 2.32·53-s − 7.01·55-s + ⋯ |
| L(s) = 1 | + 0.188·3-s − 1.04·5-s + 1.22·7-s − 0.964·9-s + 0.909·11-s − 1.09·13-s − 0.195·15-s + 0.485·17-s + 0.0747·19-s + 0.230·21-s + 0.340·23-s + 0.0819·25-s − 0.369·27-s − 1.43·29-s + 0.325·31-s + 0.171·33-s − 1.27·35-s − 1.48·37-s − 0.205·39-s + 0.193·41-s + 1.47·43-s + 1.00·45-s + 0.0261·47-s + 0.501·49-s + 0.0912·51-s − 0.319·53-s − 0.946·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 0.325T + 3T^{2} \) |
| 5 | \( 1 + 2.32T + 5T^{2} \) |
| 7 | \( 1 - 3.24T + 7T^{2} \) |
| 11 | \( 1 - 3.01T + 11T^{2} \) |
| 13 | \( 1 + 3.93T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 0.325T + 19T^{2} \) |
| 23 | \( 1 - 1.63T + 23T^{2} \) |
| 29 | \( 1 + 7.70T + 29T^{2} \) |
| 31 | \( 1 - 1.81T + 31T^{2} \) |
| 37 | \( 1 + 9.01T + 37T^{2} \) |
| 41 | \( 1 - 1.24T + 41T^{2} \) |
| 43 | \( 1 - 9.68T + 43T^{2} \) |
| 47 | \( 1 - 0.179T + 47T^{2} \) |
| 53 | \( 1 + 2.32T + 53T^{2} \) |
| 61 | \( 1 + 5.44T + 61T^{2} \) |
| 67 | \( 1 - 1.71T + 67T^{2} \) |
| 71 | \( 1 - 5.30T + 71T^{2} \) |
| 73 | \( 1 + 3.08T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 + 6.58T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 + 0.0523T + 97T^{2} \) |
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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
−7.942821756185463165028075384614, −7.66364431602985451754275908935, −6.87844018739020073448330424021, −5.73475713385978404099003033665, −5.08349578108479825449264957806, −4.24350631000400407088383647825, −3.54504558436981781234432575584, −2.52978110868445016327258229438, −1.44071524216107327808010484544, 0,
1.44071524216107327808010484544, 2.52978110868445016327258229438, 3.54504558436981781234432575584, 4.24350631000400407088383647825, 5.08349578108479825449264957806, 5.73475713385978404099003033665, 6.87844018739020073448330424021, 7.66364431602985451754275908935, 7.942821756185463165028075384614
