| L(s) = 1 | − 2.17·2-s + 3-s + 2.71·4-s − 0.590·5-s − 2.17·6-s − 3.96·7-s − 1.55·8-s + 9-s + 1.28·10-s + 6.46·11-s + 2.71·12-s − 1.30·13-s + 8.60·14-s − 0.590·15-s − 2.05·16-s + 17-s − 2.17·18-s − 5.13·19-s − 1.60·20-s − 3.96·21-s − 14.0·22-s + 1.75·23-s − 1.55·24-s − 4.65·25-s + 2.83·26-s + 27-s − 10.7·28-s + ⋯ |
| L(s) = 1 | − 1.53·2-s + 0.577·3-s + 1.35·4-s − 0.263·5-s − 0.886·6-s − 1.49·7-s − 0.548·8-s + 0.333·9-s + 0.405·10-s + 1.94·11-s + 0.783·12-s − 0.362·13-s + 2.30·14-s − 0.152·15-s − 0.514·16-s + 0.242·17-s − 0.511·18-s − 1.17·19-s − 0.358·20-s − 0.865·21-s − 2.99·22-s + 0.365·23-s − 0.316·24-s − 0.930·25-s + 0.556·26-s + 0.192·27-s − 2.03·28-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 | \( 1 + 2.17T + 2T^{2} \) |
| 5 | \( 1 + 0.590T + 5T^{2} \) |
| 7 | \( 1 + 3.96T + 7T^{2} \) |
| 11 | \( 1 - 6.46T + 11T^{2} \) |
| 13 | \( 1 + 1.30T + 13T^{2} \) |
| 19 | \( 1 + 5.13T + 19T^{2} \) |
| 23 | \( 1 - 1.75T + 23T^{2} \) |
| 29 | \( 1 + 7.00T + 29T^{2} \) |
| 31 | \( 1 - 4.10T + 31T^{2} \) |
| 37 | \( 1 + 0.396T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 1.45T + 43T^{2} \) |
| 47 | \( 1 + 1.88T + 47T^{2} \) |
| 53 | \( 1 - 6.90T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 + 1.01T + 67T^{2} \) |
| 71 | \( 1 - 0.00821T + 71T^{2} \) |
| 73 | \( 1 + 5.75T + 73T^{2} \) |
| 79 | \( 1 - 6.74T + 79T^{2} \) |
| 83 | \( 1 - 6.24T + 83T^{2} \) |
| 89 | \( 1 + 6.85T + 89T^{2} \) |
| 97 | \( 1 + 2.37T + 97T^{2} \) |
| 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
−7.63928686050592391317145631440, −6.86815308607429538197064034005, −6.57318313057584677263997548640, −5.79643698501455311161032547362, −4.21196717846296949037954975084, −3.90004043124667532713981979118, −2.88068840742416917726956404305, −2.02724983919677977956799568367, −1.06569003147134155590231744800, 0,
1.06569003147134155590231744800, 2.02724983919677977956799568367, 2.88068840742416917726956404305, 3.90004043124667532713981979118, 4.21196717846296949037954975084, 5.79643698501455311161032547362, 6.57318313057584677263997548640, 6.86815308607429538197064034005, 7.63928686050592391317145631440
