| L(s) = 1 | + 27.9·3-s − 64.0·5-s + 65.8·7-s + 539.·9-s + 635.·11-s + 467.·13-s − 1.79e3·15-s + 522.·17-s − 361·19-s + 1.84e3·21-s − 3.22e3·23-s + 981.·25-s + 8.30e3·27-s + 6.97e3·29-s − 3.38e3·31-s + 1.77e4·33-s − 4.21e3·35-s − 1.34e4·37-s + 1.30e4·39-s + 7.10e3·41-s − 1.40e4·43-s − 3.46e4·45-s − 1.44e3·47-s − 1.24e4·49-s + 1.46e4·51-s − 3.71e4·53-s − 4.07e4·55-s + ⋯ |
| L(s) = 1 | + 1.79·3-s − 1.14·5-s + 0.507·7-s + 2.22·9-s + 1.58·11-s + 0.767·13-s − 2.05·15-s + 0.438·17-s − 0.229·19-s + 0.911·21-s − 1.26·23-s + 0.313·25-s + 2.19·27-s + 1.54·29-s − 0.633·31-s + 2.84·33-s − 0.581·35-s − 1.61·37-s + 1.37·39-s + 0.660·41-s − 1.15·43-s − 2.54·45-s − 0.0953·47-s − 0.742·49-s + 0.786·51-s − 1.81·53-s − 1.81·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.053833143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.053833143\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 361T \) |
| good | 3 | \( 1 - 27.9T + 243T^{2} \) |
| 5 | \( 1 + 64.0T + 3.12e3T^{2} \) |
| 7 | \( 1 - 65.8T + 1.68e4T^{2} \) |
| 11 | \( 1 - 635.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 467.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 522.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 3.22e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.97e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.38e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.34e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.10e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.40e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.44e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.71e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.78e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.96e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.10e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 9.09e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.99e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.32e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.07e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.28e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.11e5T + 8.58e9T^{2} \) |
| 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
−13.99073734783949327876782061188, −12.49292016485261935304211822347, −11.46846883695583408908852622592, −9.862517792953791019186471355687, −8.596283517633568583178472943800, −8.121656609414142602846882040521, −6.84120663752561090423812078603, −4.19606431572491288110441971294, −3.45441998824108242666951653188, −1.56753379389982755416183322392,
1.56753379389982755416183322392, 3.45441998824108242666951653188, 4.19606431572491288110441971294, 6.84120663752561090423812078603, 8.121656609414142602846882040521, 8.596283517633568583178472943800, 9.862517792953791019186471355687, 11.46846883695583408908852622592, 12.49292016485261935304211822347, 13.99073734783949327876782061188
