| L(s) = 1 | + 1.33·2-s − 1.40·3-s − 0.214·4-s − 1.87·6-s + 0.850·7-s − 2.95·8-s − 1.02·9-s − 2.89·11-s + 0.301·12-s + 6.34·13-s + 1.13·14-s − 3.52·16-s + 4.06·17-s − 1.36·18-s − 4.69·19-s − 1.19·21-s − 3.86·22-s + 0.545·23-s + 4.15·24-s + 8.47·26-s + 5.65·27-s − 0.182·28-s − 9.52·29-s + 7.52·31-s + 1.20·32-s + 4.06·33-s + 5.42·34-s + ⋯ |
| L(s) = 1 | + 0.944·2-s − 0.811·3-s − 0.107·4-s − 0.766·6-s + 0.321·7-s − 1.04·8-s − 0.341·9-s − 0.871·11-s + 0.0870·12-s + 1.75·13-s + 0.303·14-s − 0.881·16-s + 0.985·17-s − 0.322·18-s − 1.07·19-s − 0.260·21-s − 0.823·22-s + 0.113·23-s + 0.848·24-s + 1.66·26-s + 1.08·27-s − 0.0344·28-s − 1.76·29-s + 1.35·31-s + 0.213·32-s + 0.707·33-s + 0.930·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 - 1.33T + 2T^{2} \) |
| 3 | \( 1 + 1.40T + 3T^{2} \) |
| 7 | \( 1 - 0.850T + 7T^{2} \) |
| 11 | \( 1 + 2.89T + 11T^{2} \) |
| 13 | \( 1 - 6.34T + 13T^{2} \) |
| 17 | \( 1 - 4.06T + 17T^{2} \) |
| 19 | \( 1 + 4.69T + 19T^{2} \) |
| 23 | \( 1 - 0.545T + 23T^{2} \) |
| 29 | \( 1 + 9.52T + 29T^{2} \) |
| 31 | \( 1 - 7.52T + 31T^{2} \) |
| 37 | \( 1 - 8.64T + 37T^{2} \) |
| 41 | \( 1 - 8.00T + 41T^{2} \) |
| 43 | \( 1 - 1.81T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 3.18T + 53T^{2} \) |
| 59 | \( 1 + 9.30T + 59T^{2} \) |
| 61 | \( 1 - 7.87T + 61T^{2} \) |
| 67 | \( 1 + 4.22T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 + 0.993T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 + 5.66T + 83T^{2} \) |
| 89 | \( 1 - 9.01T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−7.88806202450084523036699595939, −6.55345672595539800435873837260, −6.00605067188735448835839930429, −5.65796302197079162934118337443, −4.87166159797160610438658715789, −4.21273703657482554100386095750, −3.39361785997911458039897282083, −2.61878832213389191954245228350, −1.21569264436802114759069036768, 0,
1.21569264436802114759069036768, 2.61878832213389191954245228350, 3.39361785997911458039897282083, 4.21273703657482554100386095750, 4.87166159797160610438658715789, 5.65796302197079162934118337443, 6.00605067188735448835839930429, 6.55345672595539800435873837260, 7.88806202450084523036699595939
