| L(s) = 1 | − 0.757·2-s − 2.32·3-s − 1.42·4-s + 5-s + 1.76·6-s − 3.79·7-s + 2.59·8-s + 2.42·9-s − 0.757·10-s − 0.891·11-s + 3.32·12-s + 5.68·13-s + 2.86·14-s − 2.32·15-s + 0.889·16-s − 7.56·17-s − 1.83·18-s − 5.02·19-s − 1.42·20-s + 8.83·21-s + 0.674·22-s + 1.01·23-s − 6.04·24-s + 25-s − 4.30·26-s + 1.33·27-s + 5.40·28-s + ⋯ |
| L(s) = 1 | − 0.535·2-s − 1.34·3-s − 0.713·4-s + 0.447·5-s + 0.720·6-s − 1.43·7-s + 0.917·8-s + 0.809·9-s − 0.239·10-s − 0.268·11-s + 0.959·12-s + 1.57·13-s + 0.767·14-s − 0.601·15-s + 0.222·16-s − 1.83·17-s − 0.433·18-s − 1.15·19-s − 0.319·20-s + 1.92·21-s + 0.143·22-s + 0.212·23-s − 1.23·24-s + 0.200·25-s − 0.843·26-s + 0.256·27-s + 1.02·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 0.757T + 2T^{2} \) |
| 3 | \( 1 + 2.32T + 3T^{2} \) |
| 7 | \( 1 + 3.79T + 7T^{2} \) |
| 11 | \( 1 + 0.891T + 11T^{2} \) |
| 13 | \( 1 - 5.68T + 13T^{2} \) |
| 17 | \( 1 + 7.56T + 17T^{2} \) |
| 19 | \( 1 + 5.02T + 19T^{2} \) |
| 23 | \( 1 - 1.01T + 23T^{2} \) |
| 29 | \( 1 - 8.57T + 29T^{2} \) |
| 31 | \( 1 - 3.75T + 31T^{2} \) |
| 37 | \( 1 - 4.16T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 - 2.23T + 43T^{2} \) |
| 47 | \( 1 + 1.93T + 47T^{2} \) |
| 53 | \( 1 + 5.59T + 53T^{2} \) |
| 59 | \( 1 - 6.78T + 59T^{2} \) |
| 61 | \( 1 + 4.06T + 61T^{2} \) |
| 67 | \( 1 - 7.67T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 + 8.62T + 73T^{2} \) |
| 79 | \( 1 - 1.03T + 79T^{2} \) |
| 83 | \( 1 + 5.05T + 83T^{2} \) |
| 89 | \( 1 - 6.55T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−8.887014513655272281756988839632, −8.257016757071839627783925128554, −6.83244908048094113173545012709, −6.31849586169148011714390827877, −5.87120291105777402910062842903, −4.66763079557157910086330111487, −4.08833820261219144045891314140, −2.67926867003985908388057853223, −1.02818124120917139052882767829, 0,
1.02818124120917139052882767829, 2.67926867003985908388057853223, 4.08833820261219144045891314140, 4.66763079557157910086330111487, 5.87120291105777402910062842903, 6.31849586169148011714390827877, 6.83244908048094113173545012709, 8.257016757071839627783925128554, 8.887014513655272281756988839632
