| L(s) = 1 | + 1.38·2-s + 2.64·3-s − 0.0729·4-s − 3.88·5-s + 3.66·6-s − 3.47·7-s − 2.87·8-s + 3.98·9-s − 5.39·10-s − 1.76·11-s − 0.192·12-s − 2.35·13-s − 4.82·14-s − 10.2·15-s − 3.84·16-s + 17-s + 5.53·18-s + 6.86·19-s + 0.283·20-s − 9.18·21-s − 2.44·22-s − 8.87·23-s − 7.60·24-s + 10.0·25-s − 3.27·26-s + 2.60·27-s + 0.253·28-s + ⋯ |
| L(s) = 1 | + 0.981·2-s + 1.52·3-s − 0.0364·4-s − 1.73·5-s + 1.49·6-s − 1.31·7-s − 1.01·8-s + 1.32·9-s − 1.70·10-s − 0.531·11-s − 0.0556·12-s − 0.653·13-s − 1.28·14-s − 2.65·15-s − 0.962·16-s + 0.242·17-s + 1.30·18-s + 1.57·19-s + 0.0633·20-s − 2.00·21-s − 0.521·22-s − 1.85·23-s − 1.55·24-s + 2.01·25-s − 0.641·26-s + 0.500·27-s + 0.0478·28-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 | \( 1 - 1.38T + 2T^{2} \) |
| 3 | \( 1 - 2.64T + 3T^{2} \) |
| 5 | \( 1 + 3.88T + 5T^{2} \) |
| 7 | \( 1 + 3.47T + 7T^{2} \) |
| 11 | \( 1 + 1.76T + 11T^{2} \) |
| 13 | \( 1 + 2.35T + 13T^{2} \) |
| 19 | \( 1 - 6.86T + 19T^{2} \) |
| 23 | \( 1 + 8.87T + 23T^{2} \) |
| 29 | \( 1 + 3.50T + 29T^{2} \) |
| 31 | \( 1 - 2.45T + 31T^{2} \) |
| 37 | \( 1 - 7.99T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 - 3.16T + 43T^{2} \) |
| 47 | \( 1 + 3.14T + 47T^{2} \) |
| 53 | \( 1 - 2.21T + 53T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 15.1T + 67T^{2} \) |
| 71 | \( 1 - 6.46T + 71T^{2} \) |
| 73 | \( 1 - 1.68T + 73T^{2} \) |
| 79 | \( 1 + 6.01T + 79T^{2} \) |
| 83 | \( 1 - 3.82T + 83T^{2} \) |
| 89 | \( 1 + 7.63T + 89T^{2} \) |
| 97 | \( 1 - 1.66T + 97T^{2} \) |
| 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
−9.519028637134846975811537670682, −8.621330441622860437402397266889, −7.78857644762522548888651700148, −7.34449507040312523914628009737, −6.06883427414872357202570711389, −4.77747565669086586156196788178, −3.85133332747445574770913011190, −3.35493747076139992185090853013, −2.72353120612621954446739849017, 0,
2.72353120612621954446739849017, 3.35493747076139992185090853013, 3.85133332747445574770913011190, 4.77747565669086586156196788178, 6.06883427414872357202570711389, 7.34449507040312523914628009737, 7.78857644762522548888651700148, 8.621330441622860437402397266889, 9.519028637134846975811537670682
