| L(s) = 1 | − 2.66·2-s − 1.96·3-s + 5.11·4-s − 2.76·5-s + 5.23·6-s + 2.50·7-s − 8.29·8-s + 0.857·9-s + 7.37·10-s − 4.45·11-s − 10.0·12-s − 2.71·13-s − 6.67·14-s + 5.43·15-s + 11.8·16-s − 6.37·17-s − 2.28·18-s + 19-s − 14.1·20-s − 4.91·21-s + 11.8·22-s + 4.98·23-s + 16.2·24-s + 2.64·25-s + 7.22·26-s + 4.20·27-s + 12.7·28-s + ⋯ |
| L(s) = 1 | − 1.88·2-s − 1.13·3-s + 2.55·4-s − 1.23·5-s + 2.13·6-s + 0.945·7-s − 2.93·8-s + 0.285·9-s + 2.33·10-s − 1.34·11-s − 2.89·12-s − 0.751·13-s − 1.78·14-s + 1.40·15-s + 2.97·16-s − 1.54·17-s − 0.538·18-s + 0.229·19-s − 3.15·20-s − 1.07·21-s + 2.53·22-s + 1.04·23-s + 3.32·24-s + 0.528·25-s + 1.41·26-s + 0.809·27-s + 2.41·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 2 | \( 1 + 2.66T + 2T^{2} \) |
| 3 | \( 1 + 1.96T + 3T^{2} \) |
| 5 | \( 1 + 2.76T + 5T^{2} \) |
| 7 | \( 1 - 2.50T + 7T^{2} \) |
| 11 | \( 1 + 4.45T + 11T^{2} \) |
| 13 | \( 1 + 2.71T + 13T^{2} \) |
| 17 | \( 1 + 6.37T + 17T^{2} \) |
| 23 | \( 1 - 4.98T + 23T^{2} \) |
| 29 | \( 1 - 6.14T + 29T^{2} \) |
| 31 | \( 1 + 4.08T + 31T^{2} \) |
| 37 | \( 1 + 0.469T + 37T^{2} \) |
| 41 | \( 1 + 6.42T + 41T^{2} \) |
| 43 | \( 1 - 9.76T + 43T^{2} \) |
| 47 | \( 1 - 13.1T + 47T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 - 8.81T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 - 5.32T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 - 2.77T + 73T^{2} \) |
| 79 | \( 1 - 4.85T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 - 6.53T + 89T^{2} \) |
| 97 | \( 1 - 8.97T + 97T^{2} \) |
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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
−8.113358754820247166141541066844, −7.45853748770901399513957602062, −7.05485678935525354451380306036, −6.14388080125945713967668865446, −5.13205433352554440066383234118, −4.54964365678768302614310579392, −2.97763215824120928439466361285, −2.12936691288108114081086015983, −0.76772656808138340078504400697, 0,
0.76772656808138340078504400697, 2.12936691288108114081086015983, 2.97763215824120928439466361285, 4.54964365678768302614310579392, 5.13205433352554440066383234118, 6.14388080125945713967668865446, 7.05485678935525354451380306036, 7.45853748770901399513957602062, 8.113358754820247166141541066844
