| L(s) = 1 | + 3-s − 5-s + 1.17·7-s + 9-s − 5.41·11-s − 1.17·13-s − 15-s + 7.12·17-s − 4·19-s + 1.17·21-s − 6.97·23-s + 25-s + 27-s + 9.21·29-s − 31-s − 5.41·33-s − 1.17·35-s − 2.09·37-s − 1.17·39-s − 5.26·41-s − 6·43-s − 45-s − 1.52·47-s − 5.63·49-s + 7.12·51-s + 0.474·53-s + 5.41·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.442·7-s + 0.333·9-s − 1.63·11-s − 0.324·13-s − 0.258·15-s + 1.72·17-s − 0.917·19-s + 0.255·21-s − 1.45·23-s + 0.200·25-s + 0.192·27-s + 1.71·29-s − 0.179·31-s − 0.943·33-s − 0.197·35-s − 0.343·37-s − 0.187·39-s − 0.821·41-s − 0.914·43-s − 0.149·45-s − 0.222·47-s − 0.804·49-s + 0.998·51-s + 0.0651·53-s + 0.730·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 - 1.17T + 7T^{2} \) |
| 11 | \( 1 + 5.41T + 11T^{2} \) |
| 13 | \( 1 + 1.17T + 13T^{2} \) |
| 17 | \( 1 - 7.12T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 6.97T + 23T^{2} \) |
| 29 | \( 1 - 9.21T + 29T^{2} \) |
| 37 | \( 1 + 2.09T + 37T^{2} \) |
| 41 | \( 1 + 5.26T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + 1.52T + 47T^{2} \) |
| 53 | \( 1 - 0.474T + 53T^{2} \) |
| 59 | \( 1 - 14.6T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 + 8.14T + 71T^{2} \) |
| 73 | \( 1 + 8.74T + 73T^{2} \) |
| 79 | \( 1 - 9.46T + 79T^{2} \) |
| 83 | \( 1 + 3.86T + 83T^{2} \) |
| 89 | \( 1 + 2.04T + 89T^{2} \) |
| 97 | \( 1 + 2.73T + 97T^{2} \) |
| 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
−8.114646485763030911932291808723, −7.72469765517846670890348381809, −6.84299963518229933820177941305, −5.80821249262585016622532765894, −5.04621661972978268395173959439, −4.33514265429867354497651980505, −3.30664386234948116571506226365, −2.62610838599985250437001367294, −1.56698568788024543814463334561, 0,
1.56698568788024543814463334561, 2.62610838599985250437001367294, 3.30664386234948116571506226365, 4.33514265429867354497651980505, 5.04621661972978268395173959439, 5.80821249262585016622532765894, 6.84299963518229933820177941305, 7.72469765517846670890348381809, 8.114646485763030911932291808723
