L(s) = 1 | − 2.14·2-s + 2.61·4-s − 0.492·5-s − 0.251·7-s − 1.32·8-s + 1.05·10-s + 3.24·11-s + 4.64·13-s + 0.539·14-s − 2.38·16-s + 3.90·17-s − 1.90·19-s − 1.28·20-s − 6.97·22-s − 8.28·23-s − 4.75·25-s − 9.99·26-s − 0.657·28-s + 9.34·29-s − 0.920·31-s + 7.77·32-s − 8.39·34-s + 0.123·35-s + 8.55·37-s + 4.09·38-s + 0.654·40-s + 2.98·41-s + ⋯ |
L(s) = 1 | − 1.51·2-s + 1.30·4-s − 0.220·5-s − 0.0949·7-s − 0.469·8-s + 0.334·10-s + 0.978·11-s + 1.28·13-s + 0.144·14-s − 0.595·16-s + 0.947·17-s − 0.437·19-s − 0.288·20-s − 1.48·22-s − 1.72·23-s − 0.951·25-s − 1.95·26-s − 0.124·28-s + 1.73·29-s − 0.165·31-s + 1.37·32-s − 1.43·34-s + 0.0209·35-s + 1.40·37-s + 0.664·38-s + 0.103·40-s + 0.465·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7767074755\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7767074755\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 127 | \( 1 + T \) |
good | 2 | \( 1 + 2.14T + 2T^{2} \) |
| 5 | \( 1 + 0.492T + 5T^{2} \) |
| 7 | \( 1 + 0.251T + 7T^{2} \) |
| 11 | \( 1 - 3.24T + 11T^{2} \) |
| 13 | \( 1 - 4.64T + 13T^{2} \) |
| 17 | \( 1 - 3.90T + 17T^{2} \) |
| 19 | \( 1 + 1.90T + 19T^{2} \) |
| 23 | \( 1 + 8.28T + 23T^{2} \) |
| 29 | \( 1 - 9.34T + 29T^{2} \) |
| 31 | \( 1 + 0.920T + 31T^{2} \) |
| 37 | \( 1 - 8.55T + 37T^{2} \) |
| 41 | \( 1 - 2.98T + 41T^{2} \) |
| 43 | \( 1 - 1.41T + 43T^{2} \) |
| 47 | \( 1 + 7.43T + 47T^{2} \) |
| 53 | \( 1 - 1.05T + 53T^{2} \) |
| 59 | \( 1 + 6.89T + 59T^{2} \) |
| 61 | \( 1 + 5.54T + 61T^{2} \) |
| 67 | \( 1 - 9.13T + 67T^{2} \) |
| 71 | \( 1 - 4.84T + 71T^{2} \) |
| 73 | \( 1 - 8.54T + 73T^{2} \) |
| 79 | \( 1 - 2.23T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 3.28T + 89T^{2} \) |
| 97 | \( 1 - 4.95T + 97T^{2} \) |
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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
−9.744843422018669433869248897095, −9.015677439291806836283864032109, −8.100452612193239722046726974616, −7.85131121064894859254466145708, −6.49129197420165487520674461227, −6.11329827360719821420138915438, −4.44431882566668916181186628737, −3.49399152670404365856450443912, −1.94889105681256876502926206239, −0.872286466088419939309896601757,
0.872286466088419939309896601757, 1.94889105681256876502926206239, 3.49399152670404365856450443912, 4.44431882566668916181186628737, 6.11329827360719821420138915438, 6.49129197420165487520674461227, 7.85131121064894859254466145708, 8.100452612193239722046726974616, 9.015677439291806836283864032109, 9.744843422018669433869248897095