| L(s) = 1 | − 2.62·2-s + 3-s + 4.90·4-s + 1.25·5-s − 2.62·6-s − 3.99·7-s − 7.62·8-s + 9-s − 3.28·10-s − 1.29·11-s + 4.90·12-s − 1.60·13-s + 10.4·14-s + 1.25·15-s + 10.2·16-s − 2.81·17-s − 2.62·18-s + 0.251·19-s + 6.13·20-s − 3.99·21-s + 3.41·22-s + 3.99·23-s − 7.62·24-s − 3.43·25-s + 4.21·26-s + 27-s − 19.5·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 0.577·3-s + 2.45·4-s + 0.559·5-s − 1.07·6-s − 1.50·7-s − 2.69·8-s + 0.333·9-s − 1.03·10-s − 0.391·11-s + 1.41·12-s − 0.444·13-s + 2.80·14-s + 0.323·15-s + 2.55·16-s − 0.683·17-s − 0.619·18-s + 0.0577·19-s + 1.37·20-s − 0.871·21-s + 0.727·22-s + 0.833·23-s − 1.55·24-s − 0.686·25-s + 0.825·26-s + 0.192·27-s − 3.69·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2677 | \( 1 - T \) |
| good | 2 | \( 1 + 2.62T + 2T^{2} \) |
| 5 | \( 1 - 1.25T + 5T^{2} \) |
| 7 | \( 1 + 3.99T + 7T^{2} \) |
| 11 | \( 1 + 1.29T + 11T^{2} \) |
| 13 | \( 1 + 1.60T + 13T^{2} \) |
| 17 | \( 1 + 2.81T + 17T^{2} \) |
| 19 | \( 1 - 0.251T + 19T^{2} \) |
| 23 | \( 1 - 3.99T + 23T^{2} \) |
| 29 | \( 1 + 0.920T + 29T^{2} \) |
| 31 | \( 1 - 3.81T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 - 1.70T + 41T^{2} \) |
| 43 | \( 1 - 1.07T + 43T^{2} \) |
| 47 | \( 1 - 5.55T + 47T^{2} \) |
| 53 | \( 1 - 5.82T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 3.28T + 67T^{2} \) |
| 71 | \( 1 + 2.16T + 71T^{2} \) |
| 73 | \( 1 + 8.82T + 73T^{2} \) |
| 79 | \( 1 + 2.86T + 79T^{2} \) |
| 83 | \( 1 - 8.32T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−7.49399119301056256405053831132, −7.18072625676946997129723727168, −6.16632725207616177643313877020, −6.08319749939348061307264468198, −4.63504291658680763121113577618, −3.41760161025273964822167141179, −2.65524148799627795491683775049, −2.24240132135673914012218635561, −1.04273498004651090791688846366, 0,
1.04273498004651090791688846366, 2.24240132135673914012218635561, 2.65524148799627795491683775049, 3.41760161025273964822167141179, 4.63504291658680763121113577618, 6.08319749939348061307264468198, 6.16632725207616177643313877020, 7.18072625676946997129723727168, 7.49399119301056256405053831132
