| L(s) = 1 | − 2.65·2-s − 2.99·3-s + 5.02·4-s − 3.73·5-s + 7.93·6-s + 2.50·7-s − 8.03·8-s + 5.95·9-s + 9.90·10-s − 3.72·11-s − 15.0·12-s − 5.25·13-s − 6.65·14-s + 11.1·15-s + 11.2·16-s − 17-s − 15.7·18-s − 0.828·19-s − 18.7·20-s − 7.51·21-s + 9.88·22-s + 5.33·23-s + 24.0·24-s + 8.95·25-s + 13.9·26-s − 8.85·27-s + 12.6·28-s + ⋯ |
| L(s) = 1 | − 1.87·2-s − 1.72·3-s + 2.51·4-s − 1.67·5-s + 3.23·6-s + 0.948·7-s − 2.83·8-s + 1.98·9-s + 3.13·10-s − 1.12·11-s − 4.34·12-s − 1.45·13-s − 1.77·14-s + 2.88·15-s + 2.80·16-s − 0.242·17-s − 3.72·18-s − 0.190·19-s − 4.20·20-s − 1.63·21-s + 2.10·22-s + 1.11·23-s + 4.90·24-s + 1.79·25-s + 2.73·26-s − 1.70·27-s + 2.38·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 2 | \( 1 + 2.65T + 2T^{2} \) |
| 3 | \( 1 + 2.99T + 3T^{2} \) |
| 5 | \( 1 + 3.73T + 5T^{2} \) |
| 7 | \( 1 - 2.50T + 7T^{2} \) |
| 11 | \( 1 + 3.72T + 11T^{2} \) |
| 13 | \( 1 + 5.25T + 13T^{2} \) |
| 19 | \( 1 + 0.828T + 19T^{2} \) |
| 23 | \( 1 - 5.33T + 23T^{2} \) |
| 29 | \( 1 - 9.11T + 29T^{2} \) |
| 31 | \( 1 - 4.22T + 31T^{2} \) |
| 37 | \( 1 - 2.72T + 37T^{2} \) |
| 41 | \( 1 - 3.48T + 41T^{2} \) |
| 43 | \( 1 + 6.71T + 43T^{2} \) |
| 47 | \( 1 + 7.41T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 - 5.88T + 67T^{2} \) |
| 71 | \( 1 - 3.45T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 - 7.61T + 79T^{2} \) |
| 83 | \( 1 - 5.81T + 83T^{2} \) |
| 89 | \( 1 + 5.73T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
| 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
−9.842612323989518493412722016203, −8.428409737013514693597740343396, −7.965211810038499812593055594850, −7.17641970066835615904211118186, −6.69346521518415398729375475365, −5.20225571624356571662869082044, −4.58463740652363070094930211807, −2.64455053347707709703476367460, −0.937326673488613036075684872006, 0,
0.937326673488613036075684872006, 2.64455053347707709703476367460, 4.58463740652363070094930211807, 5.20225571624356571662869082044, 6.69346521518415398729375475365, 7.17641970066835615904211118186, 7.965211810038499812593055594850, 8.428409737013514693597740343396, 9.842612323989518493412722016203
