L(s) = 1 | − 0.0648·2-s − 3-s − 1.99·4-s + 0.118·5-s + 0.0648·6-s − 4.97·7-s + 0.259·8-s + 9-s − 0.00769·10-s − 4.34·11-s + 1.99·12-s + 3.46·13-s + 0.322·14-s − 0.118·15-s + 3.97·16-s − 17-s − 0.0648·18-s − 0.806·19-s − 0.236·20-s + 4.97·21-s + 0.281·22-s + 5.73·23-s − 0.259·24-s − 4.98·25-s − 0.224·26-s − 27-s + 9.92·28-s + ⋯ |
L(s) = 1 | − 0.0458·2-s − 0.577·3-s − 0.997·4-s + 0.0530·5-s + 0.0264·6-s − 1.87·7-s + 0.0916·8-s + 0.333·9-s − 0.00243·10-s − 1.30·11-s + 0.576·12-s + 0.960·13-s + 0.0862·14-s − 0.0306·15-s + 0.993·16-s − 0.242·17-s − 0.0152·18-s − 0.185·19-s − 0.0529·20-s + 1.08·21-s + 0.0600·22-s + 1.19·23-s − 0.0529·24-s − 0.997·25-s − 0.0440·26-s − 0.192·27-s + 1.87·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 + T \) |
good | 2 | \( 1 + 0.0648T + 2T^{2} \) |
| 5 | \( 1 - 0.118T + 5T^{2} \) |
| 7 | \( 1 + 4.97T + 7T^{2} \) |
| 11 | \( 1 + 4.34T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 19 | \( 1 + 0.806T + 19T^{2} \) |
| 23 | \( 1 - 5.73T + 23T^{2} \) |
| 29 | \( 1 + 6.51T + 29T^{2} \) |
| 31 | \( 1 + 2.26T + 31T^{2} \) |
| 37 | \( 1 + 2.01T + 37T^{2} \) |
| 41 | \( 1 - 6.33T + 41T^{2} \) |
| 43 | \( 1 - 3.21T + 43T^{2} \) |
| 47 | \( 1 - 9.60T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 2.31T + 59T^{2} \) |
| 61 | \( 1 + 7.66T + 61T^{2} \) |
| 67 | \( 1 - 8.05T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 - 5.38T + 73T^{2} \) |
| 79 | \( 1 - 9.95T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 + 8.31T + 97T^{2} \) |
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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
−7.47226091996315292978661422647, −6.71847497421561006306707364481, −5.87322288463603780077606694764, −5.62551167176792435559840539443, −4.71992654339974940213200008428, −3.78098056579282790596196353251, −3.36605407930863125148048618794, −2.31494571444002562185816633792, −0.798614337122434857747102607499, 0,
0.798614337122434857747102607499, 2.31494571444002562185816633792, 3.36605407930863125148048618794, 3.78098056579282790596196353251, 4.71992654339974940213200008428, 5.62551167176792435559840539443, 5.87322288463603780077606694764, 6.71847497421561006306707364481, 7.47226091996315292978661422647