| L(s) = 1 | + 2.74·2-s − 3-s + 5.53·4-s − 1.30·5-s − 2.74·6-s − 7-s + 9.71·8-s + 9-s − 3.58·10-s − 2.07·11-s − 5.53·12-s − 0.211·13-s − 2.74·14-s + 1.30·15-s + 15.5·16-s − 6.57·17-s + 2.74·18-s + 0.610·19-s − 7.23·20-s + 21-s − 5.68·22-s − 5.88·23-s − 9.71·24-s − 3.29·25-s − 0.580·26-s − 27-s − 5.53·28-s + ⋯ |
| L(s) = 1 | + 1.94·2-s − 0.577·3-s + 2.76·4-s − 0.584·5-s − 1.12·6-s − 0.377·7-s + 3.43·8-s + 0.333·9-s − 1.13·10-s − 0.624·11-s − 1.59·12-s − 0.0586·13-s − 0.733·14-s + 0.337·15-s + 3.89·16-s − 1.59·17-s + 0.647·18-s + 0.140·19-s − 1.61·20-s + 0.218·21-s − 1.21·22-s − 1.22·23-s − 1.98·24-s − 0.658·25-s − 0.113·26-s − 0.192·27-s − 1.04·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 - 2.74T + 2T^{2} \) |
| 5 | \( 1 + 1.30T + 5T^{2} \) |
| 11 | \( 1 + 2.07T + 11T^{2} \) |
| 13 | \( 1 + 0.211T + 13T^{2} \) |
| 17 | \( 1 + 6.57T + 17T^{2} \) |
| 19 | \( 1 - 0.610T + 19T^{2} \) |
| 23 | \( 1 + 5.88T + 23T^{2} \) |
| 29 | \( 1 - 9.95T + 29T^{2} \) |
| 31 | \( 1 + 6.19T + 31T^{2} \) |
| 37 | \( 1 + 1.14T + 37T^{2} \) |
| 41 | \( 1 + 5.38T + 41T^{2} \) |
| 43 | \( 1 + 6.20T + 43T^{2} \) |
| 47 | \( 1 - 7.82T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + 0.121T + 59T^{2} \) |
| 61 | \( 1 - 1.60T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
| 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
−7.13326383723193248961047903265, −6.51549243923158131879831104290, −6.03259231628376594423334895361, −5.26198296361978211810091639772, −4.60735473268177655243361176266, −4.10927800810283798899117142937, −3.35352581219093781459070923869, −2.52607053430802114197290351935, −1.71746209514295208517682529541, 0,
1.71746209514295208517682529541, 2.52607053430802114197290351935, 3.35352581219093781459070923869, 4.10927800810283798899117142937, 4.60735473268177655243361176266, 5.26198296361978211810091639772, 6.03259231628376594423334895361, 6.51549243923158131879831104290, 7.13326383723193248961047903265
