| L(s) = 1 | − 1.23·3-s − 2·5-s + 1.23·7-s − 1.47·9-s + 1.23·11-s + 4.47·13-s + 2.47·15-s − 6.47·19-s − 1.52·21-s + 1.23·23-s − 25-s + 5.52·27-s − 2·29-s − 1.23·31-s − 1.52·33-s − 2.47·35-s + 10.9·37-s − 5.52·39-s − 2·41-s − 1.52·43-s + 2.94·45-s + 12.9·47-s − 5.47·49-s − 2·53-s − 2.47·55-s + 8.00·57-s + 14.4·59-s + ⋯ |
| L(s) = 1 | − 0.713·3-s − 0.894·5-s + 0.467·7-s − 0.490·9-s + 0.372·11-s + 1.24·13-s + 0.638·15-s − 1.48·19-s − 0.333·21-s + 0.257·23-s − 0.200·25-s + 1.06·27-s − 0.371·29-s − 0.222·31-s − 0.265·33-s − 0.417·35-s + 1.79·37-s − 0.885·39-s − 0.312·41-s − 0.232·43-s + 0.438·45-s + 1.88·47-s − 0.781·49-s − 0.274·53-s − 0.333·55-s + 1.05·57-s + 1.88·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 1.23T + 3T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 - 1.23T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 - 1.23T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 1.23T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 1.52T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 + 6.94T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 9.23T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 1.52T + 83T^{2} \) |
| 89 | \( 1 + 7.52T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
| 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
−8.530331281265960957245710013040, −7.968654378424625052964344745597, −7.01440553160506986767525306197, −6.14932349097873625968388235514, −5.63225662549928117828236169746, −4.41998737654410291314220734517, −3.97545217050221942522207015573, −2.76966127457592119584350767723, −1.33929233782949975948073200617, 0,
1.33929233782949975948073200617, 2.76966127457592119584350767723, 3.97545217050221942522207015573, 4.41998737654410291314220734517, 5.63225662549928117828236169746, 6.14932349097873625968388235514, 7.01440553160506986767525306197, 7.968654378424625052964344745597, 8.530331281265960957245710013040
