| L(s) = 1 | − 1.54·2-s − 0.124·3-s + 0.377·4-s + 0.192·6-s + 4.01·7-s + 2.50·8-s − 2.98·9-s + 0.974·11-s − 0.0471·12-s − 2.01·13-s − 6.19·14-s − 4.61·16-s + 7.68·17-s + 4.60·18-s + 1.11·19-s − 0.502·21-s − 1.50·22-s − 2.43·23-s − 0.312·24-s + 3.11·26-s + 0.747·27-s + 1.51·28-s − 6.25·29-s + 31-s + 2.10·32-s − 0.121·33-s − 11.8·34-s + ⋯ |
| L(s) = 1 | − 1.09·2-s − 0.0721·3-s + 0.188·4-s + 0.0786·6-s + 1.51·7-s + 0.884·8-s − 0.994·9-s + 0.293·11-s − 0.0136·12-s − 0.559·13-s − 1.65·14-s − 1.15·16-s + 1.86·17-s + 1.08·18-s + 0.255·19-s − 0.109·21-s − 0.320·22-s − 0.507·23-s − 0.0638·24-s + 0.610·26-s + 0.143·27-s + 0.286·28-s − 1.16·29-s + 0.179·31-s + 0.372·32-s − 0.0211·33-s − 2.03·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8892341529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8892341529\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 1.54T + 2T^{2} \) |
| 3 | \( 1 + 0.124T + 3T^{2} \) |
| 7 | \( 1 - 4.01T + 7T^{2} \) |
| 11 | \( 1 - 0.974T + 11T^{2} \) |
| 13 | \( 1 + 2.01T + 13T^{2} \) |
| 17 | \( 1 - 7.68T + 17T^{2} \) |
| 19 | \( 1 - 1.11T + 19T^{2} \) |
| 23 | \( 1 + 2.43T + 23T^{2} \) |
| 29 | \( 1 + 6.25T + 29T^{2} \) |
| 37 | \( 1 + 9.60T + 37T^{2} \) |
| 41 | \( 1 - 8.36T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 4.09T + 53T^{2} \) |
| 59 | \( 1 - 9.85T + 59T^{2} \) |
| 61 | \( 1 + 4.17T + 61T^{2} \) |
| 67 | \( 1 + 7.41T + 67T^{2} \) |
| 71 | \( 1 - 6.02T + 71T^{2} \) |
| 73 | \( 1 - 4.20T + 73T^{2} \) |
| 79 | \( 1 + 2.19T + 79T^{2} \) |
| 83 | \( 1 - 9.70T + 83T^{2} \) |
| 89 | \( 1 - 3.96T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−10.29173978379758512228148591926, −9.290831789010529180730802858760, −8.668666995620501290451470402376, −7.71320788272486281200061350848, −7.50053109058634994087111009554, −5.74597765531176197591442367452, −5.08783591829484305782456912695, −3.87726969173245086849368973809, −2.22246928833494040906715267163, −0.969979360313896299703727184922,
0.969979360313896299703727184922, 2.22246928833494040906715267163, 3.87726969173245086849368973809, 5.08783591829484305782456912695, 5.74597765531176197591442367452, 7.50053109058634994087111009554, 7.71320788272486281200061350848, 8.668666995620501290451470402376, 9.290831789010529180730802858760, 10.29173978379758512228148591926
