| L(s) = 1 | + 2-s − 3.23·3-s + 4-s − 3.65·5-s − 3.23·6-s + 2.39·7-s + 8-s + 7.43·9-s − 3.65·10-s + 0.229·11-s − 3.23·12-s − 1.51·13-s + 2.39·14-s + 11.7·15-s + 16-s + 1.23·17-s + 7.43·18-s + 19-s − 3.65·20-s − 7.72·21-s + 0.229·22-s − 8.59·23-s − 3.23·24-s + 8.33·25-s − 1.51·26-s − 14.3·27-s + 2.39·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.86·3-s + 0.5·4-s − 1.63·5-s − 1.31·6-s + 0.903·7-s + 0.353·8-s + 2.47·9-s − 1.15·10-s + 0.0690·11-s − 0.932·12-s − 0.421·13-s + 0.638·14-s + 3.04·15-s + 0.250·16-s + 0.299·17-s + 1.75·18-s + 0.229·19-s − 0.816·20-s − 1.68·21-s + 0.0488·22-s − 1.79·23-s − 0.659·24-s + 1.66·25-s − 0.298·26-s − 2.75·27-s + 0.451·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 3.23T + 3T^{2} \) |
| 5 | \( 1 + 3.65T + 5T^{2} \) |
| 7 | \( 1 - 2.39T + 7T^{2} \) |
| 11 | \( 1 - 0.229T + 11T^{2} \) |
| 13 | \( 1 + 1.51T + 13T^{2} \) |
| 17 | \( 1 - 1.23T + 17T^{2} \) |
| 23 | \( 1 + 8.59T + 23T^{2} \) |
| 29 | \( 1 + 2.80T + 29T^{2} \) |
| 31 | \( 1 - 4.09T + 31T^{2} \) |
| 37 | \( 1 + 7.69T + 37T^{2} \) |
| 41 | \( 1 - 6.97T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 2.41T + 53T^{2} \) |
| 59 | \( 1 - 5.91T + 59T^{2} \) |
| 61 | \( 1 - 5.03T + 61T^{2} \) |
| 67 | \( 1 + 1.27T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 5.30T + 73T^{2} \) |
| 79 | \( 1 - 4.63T + 79T^{2} \) |
| 83 | \( 1 - 0.441T + 83T^{2} \) |
| 89 | \( 1 - 3.69T + 89T^{2} \) |
| 97 | \( 1 - 7.94T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.41724150179195757658549749926, −6.70838925485580000849032672283, −5.97410747567096320731572833166, −5.22496516161833556153775025827, −4.78352244487710473195868917268, −4.08282823355455037042559429058, −3.63671893774816011824181872326, −2.10324036713783251922239193351, −1.00179035525224963913414570963, 0,
1.00179035525224963913414570963, 2.10324036713783251922239193351, 3.63671893774816011824181872326, 4.08282823355455037042559429058, 4.78352244487710473195868917268, 5.22496516161833556153775025827, 5.97410747567096320731572833166, 6.70838925485580000849032672283, 7.41724150179195757658549749926
