| L(s) = 1 | + 5-s − 0.632·7-s + 5.48·11-s + 6.85·13-s − 5.48·17-s + 1.36·19-s − 0.367·23-s + 25-s + 1.63·29-s − 7.48·31-s − 0.632·35-s + 4.63·37-s + 11.7·41-s + 9.33·43-s + 4.36·47-s − 6.59·49-s + 4·53-s + 5.48·55-s − 5.26·59-s + 4.63·61-s + 6.85·65-s − 10.3·67-s − 7.70·71-s + 5.89·73-s − 3.46·77-s − 10.8·79-s − 12.4·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.239·7-s + 1.65·11-s + 1.90·13-s − 1.32·17-s + 0.313·19-s − 0.0765·23-s + 0.200·25-s + 0.303·29-s − 1.34·31-s − 0.106·35-s + 0.761·37-s + 1.82·41-s + 1.42·43-s + 0.637·47-s − 0.942·49-s + 0.549·53-s + 0.739·55-s − 0.685·59-s + 0.593·61-s + 0.849·65-s − 1.26·67-s − 0.914·71-s + 0.690·73-s − 0.395·77-s − 1.22·79-s − 1.36·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.533192032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.533192032\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + 0.632T + 7T^{2} \) |
| 11 | \( 1 - 5.48T + 11T^{2} \) |
| 13 | \( 1 - 6.85T + 13T^{2} \) |
| 17 | \( 1 + 5.48T + 17T^{2} \) |
| 19 | \( 1 - 1.36T + 19T^{2} \) |
| 23 | \( 1 + 0.367T + 23T^{2} \) |
| 29 | \( 1 - 1.63T + 29T^{2} \) |
| 31 | \( 1 + 7.48T + 31T^{2} \) |
| 37 | \( 1 - 4.63T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 - 9.33T + 43T^{2} \) |
| 47 | \( 1 - 4.36T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 5.26T + 59T^{2} \) |
| 61 | \( 1 - 4.63T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 + 7.70T + 71T^{2} \) |
| 73 | \( 1 - 5.89T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 7.36T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.757322577771666803081972942152, −7.56486966101122073472382767136, −6.78332993546204918744549680249, −6.10476508166183090251754437784, −5.77752735048831616826318314213, −4.31616876992581449809646230805, −3.99415722173210591259200347202, −2.96458759878088830801267445713, −1.80043208388522360213896660480, −0.971137944634753299428757842514,
0.971137944634753299428757842514, 1.80043208388522360213896660480, 2.96458759878088830801267445713, 3.99415722173210591259200347202, 4.31616876992581449809646230805, 5.77752735048831616826318314213, 6.10476508166183090251754437784, 6.78332993546204918744549680249, 7.56486966101122073472382767136, 8.757322577771666803081972942152
