L(s) = 1 | − 2.75·2-s + 0.843·3-s + 5.60·4-s + 3.43·5-s − 2.32·6-s − 0.344·7-s − 9.95·8-s − 2.28·9-s − 9.47·10-s − 11-s + 4.72·12-s − 4.61·13-s + 0.951·14-s + 2.89·15-s + 16.2·16-s + 17-s + 6.31·18-s + 7.54·19-s + 19.2·20-s − 0.290·21-s + 2.75·22-s + 1.39·23-s − 8.38·24-s + 6.80·25-s + 12.7·26-s − 4.45·27-s − 1.93·28-s + ⋯ |
L(s) = 1 | − 1.95·2-s + 0.486·3-s + 2.80·4-s + 1.53·5-s − 0.949·6-s − 0.130·7-s − 3.51·8-s − 0.763·9-s − 2.99·10-s − 0.301·11-s + 1.36·12-s − 1.28·13-s + 0.254·14-s + 0.748·15-s + 4.05·16-s + 0.242·17-s + 1.48·18-s + 1.73·19-s + 4.30·20-s − 0.0634·21-s + 0.588·22-s + 0.289·23-s − 1.71·24-s + 1.36·25-s + 2.49·26-s − 0.858·27-s − 0.365·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
good | 2 | \( 1 + 2.75T + 2T^{2} \) |
| 3 | \( 1 - 0.843T + 3T^{2} \) |
| 5 | \( 1 - 3.43T + 5T^{2} \) |
| 7 | \( 1 + 0.344T + 7T^{2} \) |
| 13 | \( 1 + 4.61T + 13T^{2} \) |
| 19 | \( 1 - 7.54T + 19T^{2} \) |
| 23 | \( 1 - 1.39T + 23T^{2} \) |
| 29 | \( 1 + 4.70T + 29T^{2} \) |
| 31 | \( 1 - 2.92T + 31T^{2} \) |
| 37 | \( 1 - 5.72T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 14.0T + 53T^{2} \) |
| 59 | \( 1 - 3.89T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 6.76T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 + 2.55T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 17.1T + 83T^{2} \) |
| 89 | \( 1 - 2.44T + 89T^{2} \) |
| 97 | \( 1 - 6.64T + 97T^{2} \) |
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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
−7.62285685329289845130144222837, −7.16016913412874437591106710332, −6.24313973298152332147215541255, −5.72829694206480849676632803344, −5.05539078521464542476764487322, −3.08935454391427154114152109848, −2.83454866623433394437592971511, −2.00719080357993155154216705092, −1.28600674847326691290381304229, 0,
1.28600674847326691290381304229, 2.00719080357993155154216705092, 2.83454866623433394437592971511, 3.08935454391427154114152109848, 5.05539078521464542476764487322, 5.72829694206480849676632803344, 6.24313973298152332147215541255, 7.16016913412874437591106710332, 7.62285685329289845130144222837