| L(s) = 1 | − 1.61·2-s − 2·3-s + 0.618·4-s − 0.618·5-s + 3.23·6-s − 7-s + 2.23·8-s + 9-s + 1.00·10-s + 5.23·11-s − 1.23·12-s − 4.61·13-s + 1.61·14-s + 1.23·15-s − 4.85·16-s + 0.763·17-s − 1.61·18-s − 3.85·19-s − 0.381·20-s + 2·21-s − 8.47·22-s + 4.47·23-s − 4.47·24-s − 4.61·25-s + 7.47·26-s + 4·27-s − 0.618·28-s + ⋯ |
| L(s) = 1 | − 1.14·2-s − 1.15·3-s + 0.309·4-s − 0.276·5-s + 1.32·6-s − 0.377·7-s + 0.790·8-s + 0.333·9-s + 0.316·10-s + 1.57·11-s − 0.356·12-s − 1.28·13-s + 0.432·14-s + 0.319·15-s − 1.21·16-s + 0.185·17-s − 0.381·18-s − 0.884·19-s − 0.0854·20-s + 0.436·21-s − 1.80·22-s + 0.932·23-s − 0.912·24-s − 0.923·25-s + 1.46·26-s + 0.769·27-s − 0.116·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3229 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3229 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3229 | \( 1+O(T) \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 3 | \( 1 + 2T + 3T^{2} \) |
| 5 | \( 1 + 0.618T + 5T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 5.23T + 11T^{2} \) |
| 13 | \( 1 + 4.61T + 13T^{2} \) |
| 17 | \( 1 - 0.763T + 17T^{2} \) |
| 19 | \( 1 + 3.85T + 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 - 5.38T + 29T^{2} \) |
| 31 | \( 1 + 7.38T + 31T^{2} \) |
| 37 | \( 1 + 3.09T + 37T^{2} \) |
| 41 | \( 1 + 1.09T + 41T^{2} \) |
| 43 | \( 1 - 5.09T + 43T^{2} \) |
| 47 | \( 1 - 2.23T + 47T^{2} \) |
| 53 | \( 1 - 1.47T + 53T^{2} \) |
| 59 | \( 1 + 2.23T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 0.708T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 + 2.29T + 79T^{2} \) |
| 83 | \( 1 - 6.38T + 83T^{2} \) |
| 89 | \( 1 - 7.79T + 89T^{2} \) |
| 97 | \( 1 + 3.85T + 97T^{2} \) |
| 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
−8.427060992394067699502411002199, −7.49468130967194770377486496147, −6.83204308362645302425423089947, −6.30265680586406888510255250824, −5.22911614526359833785589068610, −4.53639698697072475497395211070, −3.63285882790870401122413794115, −2.14759954733567702445904590174, −0.959966070508571975900328113569, 0,
0.959966070508571975900328113569, 2.14759954733567702445904590174, 3.63285882790870401122413794115, 4.53639698697072475497395211070, 5.22911614526359833785589068610, 6.30265680586406888510255250824, 6.83204308362645302425423089947, 7.49468130967194770377486496147, 8.427060992394067699502411002199
