| L(s) = 1 | + 0.797·2-s − 3.35·3-s − 1.36·4-s − 2.67·6-s − 7-s − 2.68·8-s + 8.22·9-s + 11-s + 4.56·12-s − 4.12·13-s − 0.797·14-s + 0.585·16-s + 2.39·17-s + 6.56·18-s + 3.81·19-s + 3.35·21-s + 0.797·22-s + 5.37·23-s + 8.99·24-s − 3.29·26-s − 17.5·27-s + 1.36·28-s − 4.69·29-s + 0.178·31-s + 5.83·32-s − 3.35·33-s + 1.91·34-s + ⋯ |
| L(s) = 1 | + 0.564·2-s − 1.93·3-s − 0.681·4-s − 1.09·6-s − 0.377·7-s − 0.948·8-s + 2.74·9-s + 0.301·11-s + 1.31·12-s − 1.14·13-s − 0.213·14-s + 0.146·16-s + 0.581·17-s + 1.54·18-s + 0.875·19-s + 0.731·21-s + 0.170·22-s + 1.12·23-s + 1.83·24-s − 0.646·26-s − 3.36·27-s + 0.257·28-s − 0.871·29-s + 0.0321·31-s + 1.03·32-s − 0.583·33-s + 0.328·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 0.797T + 2T^{2} \) |
| 3 | \( 1 + 3.35T + 3T^{2} \) |
| 13 | \( 1 + 4.12T + 13T^{2} \) |
| 17 | \( 1 - 2.39T + 17T^{2} \) |
| 19 | \( 1 - 3.81T + 19T^{2} \) |
| 23 | \( 1 - 5.37T + 23T^{2} \) |
| 29 | \( 1 + 4.69T + 29T^{2} \) |
| 31 | \( 1 - 0.178T + 31T^{2} \) |
| 37 | \( 1 - 1.61T + 37T^{2} \) |
| 41 | \( 1 - 9.62T + 41T^{2} \) |
| 43 | \( 1 - 6.65T + 43T^{2} \) |
| 47 | \( 1 + 6.41T + 47T^{2} \) |
| 53 | \( 1 - 3.38T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 + 4.92T + 61T^{2} \) |
| 67 | \( 1 + 9.36T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 + 7.66T + 73T^{2} \) |
| 79 | \( 1 - 5.01T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 + 1.59T + 89T^{2} \) |
| 97 | \( 1 + 2.88T + 97T^{2} \) |
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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
−9.252359196280636924406115344639, −7.61677089870906504333375036274, −7.08663999534365320506161672522, −6.00550673127706042262628881426, −5.62618786537031290153792285245, −4.80256392747000822408099649525, −4.26678075700018869611685050324, −3.07304168334680295327685966118, −1.16038332994166718377074359118, 0,
1.16038332994166718377074359118, 3.07304168334680295327685966118, 4.26678075700018869611685050324, 4.80256392747000822408099649525, 5.62618786537031290153792285245, 6.00550673127706042262628881426, 7.08663999534365320506161672522, 7.61677089870906504333375036274, 9.252359196280636924406115344639
