| L(s) = 1 | − 2.50·2-s − 0.199·3-s + 4.27·4-s − 1.81·5-s + 0.500·6-s − 1.67·7-s − 5.70·8-s − 2.96·9-s + 4.53·10-s − 2.18·11-s − 0.853·12-s + 13-s + 4.18·14-s + 0.361·15-s + 5.73·16-s − 5.47·17-s + 7.41·18-s − 5.01·19-s − 7.74·20-s + 0.333·21-s + 5.46·22-s − 7.67·23-s + 1.13·24-s − 1.72·25-s − 2.50·26-s + 1.19·27-s − 7.14·28-s + ⋯ |
| L(s) = 1 | − 1.77·2-s − 0.115·3-s + 2.13·4-s − 0.809·5-s + 0.204·6-s − 0.631·7-s − 2.01·8-s − 0.986·9-s + 1.43·10-s − 0.657·11-s − 0.246·12-s + 0.277·13-s + 1.11·14-s + 0.0933·15-s + 1.43·16-s − 1.32·17-s + 1.74·18-s − 1.15·19-s − 1.73·20-s + 0.0727·21-s + 1.16·22-s − 1.60·23-s + 0.232·24-s − 0.344·25-s − 0.491·26-s + 0.229·27-s − 1.34·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 619 | \( 1 - T \) |
| good | 2 | \( 1 + 2.50T + 2T^{2} \) |
| 3 | \( 1 + 0.199T + 3T^{2} \) |
| 5 | \( 1 + 1.81T + 5T^{2} \) |
| 7 | \( 1 + 1.67T + 7T^{2} \) |
| 11 | \( 1 + 2.18T + 11T^{2} \) |
| 17 | \( 1 + 5.47T + 17T^{2} \) |
| 19 | \( 1 + 5.01T + 19T^{2} \) |
| 23 | \( 1 + 7.67T + 23T^{2} \) |
| 29 | \( 1 - 6.13T + 29T^{2} \) |
| 31 | \( 1 - 6.78T + 31T^{2} \) |
| 37 | \( 1 - 2.77T + 37T^{2} \) |
| 41 | \( 1 - 9.42T + 41T^{2} \) |
| 43 | \( 1 - 9.65T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 5.00T + 53T^{2} \) |
| 59 | \( 1 - 9.88T + 59T^{2} \) |
| 61 | \( 1 - 4.28T + 61T^{2} \) |
| 67 | \( 1 - 8.87T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 1.61T + 79T^{2} \) |
| 83 | \( 1 + 7.95T + 83T^{2} \) |
| 89 | \( 1 + 6.16T + 89T^{2} \) |
| 97 | \( 1 + 2.74T + 97T^{2} \) |
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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
−7.981554023321795797192579839923, −6.85003617881134770665705439803, −6.42665309396604618628352647807, −5.83003792458836543332987877812, −4.55099603920958235226879246895, −3.77010838523592797711919830534, −2.55401469681182028028708421838, −2.30031388377654875830593552105, −0.69760093964343253216004106224, 0,
0.69760093964343253216004106224, 2.30031388377654875830593552105, 2.55401469681182028028708421838, 3.77010838523592797711919830534, 4.55099603920958235226879246895, 5.83003792458836543332987877812, 6.42665309396604618628352647807, 6.85003617881134770665705439803, 7.981554023321795797192579839923
