| L(s) = 1 | − 0.791·2-s + 3-s − 1.37·4-s + 5-s − 0.791·6-s + 2.45·7-s + 2.66·8-s + 9-s − 0.791·10-s − 1.92·11-s − 1.37·12-s + 5.90·13-s − 1.94·14-s + 15-s + 0.636·16-s − 7.53·17-s − 0.791·18-s − 5.97·19-s − 1.37·20-s + 2.45·21-s + 1.52·22-s + 7.54·23-s + 2.66·24-s + 25-s − 4.66·26-s + 27-s − 3.37·28-s + ⋯ |
| L(s) = 1 | − 0.559·2-s + 0.577·3-s − 0.687·4-s + 0.447·5-s − 0.322·6-s + 0.927·7-s + 0.943·8-s + 0.333·9-s − 0.250·10-s − 0.580·11-s − 0.396·12-s + 1.63·13-s − 0.518·14-s + 0.258·15-s + 0.159·16-s − 1.82·17-s − 0.186·18-s − 1.36·19-s − 0.307·20-s + 0.535·21-s + 0.324·22-s + 1.57·23-s + 0.544·24-s + 0.200·25-s − 0.915·26-s + 0.192·27-s − 0.637·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 0.791T + 2T^{2} \) |
| 7 | \( 1 - 2.45T + 7T^{2} \) |
| 11 | \( 1 + 1.92T + 11T^{2} \) |
| 13 | \( 1 - 5.90T + 13T^{2} \) |
| 17 | \( 1 + 7.53T + 17T^{2} \) |
| 19 | \( 1 + 5.97T + 19T^{2} \) |
| 23 | \( 1 - 7.54T + 23T^{2} \) |
| 29 | \( 1 + 3.11T + 29T^{2} \) |
| 31 | \( 1 + 6.03T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 + 2.14T + 41T^{2} \) |
| 43 | \( 1 + 9.58T + 43T^{2} \) |
| 47 | \( 1 + 7.46T + 47T^{2} \) |
| 53 | \( 1 - 2.30T + 53T^{2} \) |
| 59 | \( 1 - 0.645T + 59T^{2} \) |
| 61 | \( 1 + 8.35T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 - 1.08T + 79T^{2} \) |
| 83 | \( 1 + 6.21T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 + 6.92T + 97T^{2} \) |
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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
−8.140386827145205493147183782051, −7.07682543267131784286985805354, −6.52909684650419551800227807643, −5.39206019885047818347157075260, −4.80902821505902882302790726279, −4.09542214454730285471233830736, −3.25044306273637117942555444423, −1.94033608763919899531999265746, −1.52714949573788920456459399579, 0,
1.52714949573788920456459399579, 1.94033608763919899531999265746, 3.25044306273637117942555444423, 4.09542214454730285471233830736, 4.80902821505902882302790726279, 5.39206019885047818347157075260, 6.52909684650419551800227807643, 7.07682543267131784286985805354, 8.140386827145205493147183782051
