L(s) = 1 | − 0.525·2-s + 0.0103·3-s − 1.72·4-s − 5-s − 0.00545·6-s + 7-s + 1.95·8-s − 2.99·9-s + 0.525·10-s + 2.45·11-s − 0.0178·12-s − 2.86·13-s − 0.525·14-s − 0.0103·15-s + 2.41·16-s − 1.89·17-s + 1.57·18-s + 6.96·19-s + 1.72·20-s + 0.0103·21-s − 1.29·22-s − 1.83·23-s + 0.0203·24-s + 25-s + 1.50·26-s − 0.0622·27-s − 1.72·28-s + ⋯ |
L(s) = 1 | − 0.371·2-s + 0.00599·3-s − 0.861·4-s − 0.447·5-s − 0.00222·6-s + 0.377·7-s + 0.691·8-s − 0.999·9-s + 0.166·10-s + 0.740·11-s − 0.00516·12-s − 0.793·13-s − 0.140·14-s − 0.00268·15-s + 0.604·16-s − 0.458·17-s + 0.371·18-s + 1.59·19-s + 0.385·20-s + 0.00226·21-s − 0.275·22-s − 0.382·23-s + 0.00414·24-s + 0.200·25-s + 0.294·26-s − 0.0119·27-s − 0.325·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 - T \) |
good | 2 | \( 1 + 0.525T + 2T^{2} \) |
| 3 | \( 1 - 0.0103T + 3T^{2} \) |
| 11 | \( 1 - 2.45T + 11T^{2} \) |
| 13 | \( 1 + 2.86T + 13T^{2} \) |
| 17 | \( 1 + 1.89T + 17T^{2} \) |
| 19 | \( 1 - 6.96T + 19T^{2} \) |
| 23 | \( 1 + 1.83T + 23T^{2} \) |
| 29 | \( 1 + 8.45T + 29T^{2} \) |
| 31 | \( 1 - 3.05T + 31T^{2} \) |
| 37 | \( 1 + 6.07T + 37T^{2} \) |
| 41 | \( 1 - 9.43T + 41T^{2} \) |
| 43 | \( 1 + 0.608T + 43T^{2} \) |
| 47 | \( 1 + 3.36T + 47T^{2} \) |
| 53 | \( 1 - 5.23T + 53T^{2} \) |
| 59 | \( 1 + 3.40T + 59T^{2} \) |
| 61 | \( 1 - 1.78T + 61T^{2} \) |
| 67 | \( 1 + 4.69T + 67T^{2} \) |
| 71 | \( 1 - 16.6T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 - 2.47T + 83T^{2} \) |
| 89 | \( 1 + 4.96T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−7.63568488915483664921898410356, −7.06010232360531328453321633686, −5.96801096216017282396311832432, −5.31706080574494915042207363407, −4.71546263717416519483660560906, −3.88784062473758588808443068923, −3.24945998497930772997575595894, −2.14153108605284536351932808715, −1.01462239806734352963575357624, 0,
1.01462239806734352963575357624, 2.14153108605284536351932808715, 3.24945998497930772997575595894, 3.88784062473758588808443068923, 4.71546263717416519483660560906, 5.31706080574494915042207363407, 5.96801096216017282396311832432, 7.06010232360531328453321633686, 7.63568488915483664921898410356