| L(s) = 1 | + 1.40·2-s − 3-s − 0.0266·4-s + 2.64·5-s − 1.40·6-s − 2.79·7-s − 2.84·8-s + 9-s + 3.71·10-s + 1.62·11-s + 0.0266·12-s + 0.165·13-s − 3.92·14-s − 2.64·15-s − 3.94·16-s − 17-s + 1.40·18-s − 0.0799·19-s − 0.0705·20-s + 2.79·21-s + 2.27·22-s + 4.53·23-s + 2.84·24-s + 2.01·25-s + 0.232·26-s − 27-s + 0.0744·28-s + ⋯ |
| L(s) = 1 | + 0.993·2-s − 0.577·3-s − 0.0133·4-s + 1.18·5-s − 0.573·6-s − 1.05·7-s − 1.00·8-s + 0.333·9-s + 1.17·10-s + 0.488·11-s + 0.00769·12-s + 0.0459·13-s − 1.04·14-s − 0.683·15-s − 0.986·16-s − 0.242·17-s + 0.331·18-s − 0.0183·19-s − 0.0157·20-s + 0.609·21-s + 0.485·22-s + 0.944·23-s + 0.581·24-s + 0.402·25-s + 0.0456·26-s − 0.192·27-s + 0.0140·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 - 1.40T + 2T^{2} \) |
| 5 | \( 1 - 2.64T + 5T^{2} \) |
| 7 | \( 1 + 2.79T + 7T^{2} \) |
| 11 | \( 1 - 1.62T + 11T^{2} \) |
| 13 | \( 1 - 0.165T + 13T^{2} \) |
| 19 | \( 1 + 0.0799T + 19T^{2} \) |
| 23 | \( 1 - 4.53T + 23T^{2} \) |
| 29 | \( 1 + 2.83T + 29T^{2} \) |
| 31 | \( 1 - 4.25T + 31T^{2} \) |
| 37 | \( 1 + 3.78T + 37T^{2} \) |
| 41 | \( 1 - 8.76T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 - 3.98T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 + 8.23T + 59T^{2} \) |
| 61 | \( 1 - 8.41T + 61T^{2} \) |
| 67 | \( 1 - 5.72T + 67T^{2} \) |
| 71 | \( 1 - 7.22T + 71T^{2} \) |
| 73 | \( 1 + 0.0542T + 73T^{2} \) |
| 79 | \( 1 + 4.31T + 79T^{2} \) |
| 83 | \( 1 + 7.16T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 - 2.43T + 97T^{2} \) |
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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
−6.95311596873816556244930724380, −6.55624000186437970203127648997, −6.00903686556754879530678490067, −5.43307953038293313480389902919, −4.80883433076308310918690309883, −3.97973401553010792221293533756, −3.22198956334987079783790086686, −2.47236289140882840868368422254, −1.32342276617596377272203778189, 0,
1.32342276617596377272203778189, 2.47236289140882840868368422254, 3.22198956334987079783790086686, 3.97973401553010792221293533756, 4.80883433076308310918690309883, 5.43307953038293313480389902919, 6.00903686556754879530678490067, 6.55624000186437970203127648997, 6.95311596873816556244930724380
