| L(s) = 1 | − 2.63·2-s − 0.811·3-s + 4.93·4-s + 5-s + 2.13·6-s + 5.26·7-s − 7.73·8-s − 2.34·9-s − 2.63·10-s − 11-s − 4.00·12-s − 3.11·13-s − 13.8·14-s − 0.811·15-s + 10.4·16-s − 1.87·17-s + 6.16·18-s + 0.651·19-s + 4.93·20-s − 4.27·21-s + 2.63·22-s − 2.26·23-s + 6.27·24-s + 25-s + 8.19·26-s + 4.33·27-s + 26.0·28-s + ⋯ |
| L(s) = 1 | − 1.86·2-s − 0.468·3-s + 2.46·4-s + 0.447·5-s + 0.872·6-s + 1.99·7-s − 2.73·8-s − 0.780·9-s − 0.832·10-s − 0.301·11-s − 1.15·12-s − 0.863·13-s − 3.70·14-s − 0.209·15-s + 2.62·16-s − 0.454·17-s + 1.45·18-s + 0.149·19-s + 1.10·20-s − 0.933·21-s + 0.561·22-s − 0.473·23-s + 1.28·24-s + 0.200·25-s + 1.60·26-s + 0.834·27-s + 4.91·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 + 2.63T + 2T^{2} \) |
| 3 | \( 1 + 0.811T + 3T^{2} \) |
| 7 | \( 1 - 5.26T + 7T^{2} \) |
| 13 | \( 1 + 3.11T + 13T^{2} \) |
| 17 | \( 1 + 1.87T + 17T^{2} \) |
| 19 | \( 1 - 0.651T + 19T^{2} \) |
| 23 | \( 1 + 2.26T + 23T^{2} \) |
| 29 | \( 1 - 1.91T + 29T^{2} \) |
| 31 | \( 1 + 3.24T + 31T^{2} \) |
| 37 | \( 1 + 2.93T + 37T^{2} \) |
| 41 | \( 1 + 8.89T + 41T^{2} \) |
| 43 | \( 1 - 5.24T + 43T^{2} \) |
| 47 | \( 1 - 6.50T + 47T^{2} \) |
| 53 | \( 1 - 9.03T + 53T^{2} \) |
| 59 | \( 1 + 6.80T + 59T^{2} \) |
| 61 | \( 1 + 2.23T + 61T^{2} \) |
| 67 | \( 1 + 6.16T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 79 | \( 1 + 3.39T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 3.31T + 97T^{2} \) |
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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
−8.367046515567083962484374805471, −7.47094085002345604729762993278, −7.06216143693666348437337528319, −5.91108221621249049062795122492, −5.38057929782332557228359218720, −4.48535727165949690045039218012, −2.76033884999090416197676254939, −2.06546592855561998898836408364, −1.27229016300353656698259019683, 0,
1.27229016300353656698259019683, 2.06546592855561998898836408364, 2.76033884999090416197676254939, 4.48535727165949690045039218012, 5.38057929782332557228359218720, 5.91108221621249049062795122492, 7.06216143693666348437337528319, 7.47094085002345604729762993278, 8.367046515567083962484374805471
