| L(s) = 1 | + 4.76·3-s + 18.8·5-s + 185.·7-s − 220.·9-s − 121·11-s + 450.·13-s + 89.6·15-s + 1.87e3·17-s + 271.·19-s + 885.·21-s + 2.18e3·23-s − 2.77e3·25-s − 2.20e3·27-s + 7.29e3·29-s + 5.23e3·31-s − 576.·33-s + 3.49e3·35-s + 8.12e3·37-s + 2.14e3·39-s + 1.33e4·41-s − 2.14e4·43-s − 4.14e3·45-s − 2.84e4·47-s + 1.76e4·49-s + 8.94e3·51-s − 5.60e3·53-s − 2.27e3·55-s + ⋯ |
| L(s) = 1 | + 0.305·3-s + 0.336·5-s + 1.43·7-s − 0.906·9-s − 0.301·11-s + 0.740·13-s + 0.102·15-s + 1.57·17-s + 0.172·19-s + 0.437·21-s + 0.859·23-s − 0.886·25-s − 0.582·27-s + 1.60·29-s + 0.978·31-s − 0.0921·33-s + 0.482·35-s + 0.975·37-s + 0.226·39-s + 1.23·41-s − 1.77·43-s − 0.304·45-s − 1.87·47-s + 1.05·49-s + 0.481·51-s − 0.274·53-s − 0.101·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.411566737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.411566737\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 3 | \( 1 - 4.76T + 243T^{2} \) |
| 5 | \( 1 - 18.8T + 3.12e3T^{2} \) |
| 7 | \( 1 - 185.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 450.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.87e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 271.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.18e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.29e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.23e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.12e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.33e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.14e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.84e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 5.60e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.54e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.06e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.54e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.45e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.23e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.38e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.43e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.70e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.54e3T + 8.58e9T^{2} \) |
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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
−13.46608028965734558368467201644, −11.93542752930950848670594718744, −11.14713281637946271773117000662, −9.910611902728387012086470740501, −8.474076944337982711264776410486, −7.85250401913226081527003988553, −5.99912202698253601773987535907, −4.84933421539356018805823812814, −3.00008693074558288604490723017, −1.31279967769171880461670489908,
1.31279967769171880461670489908, 3.00008693074558288604490723017, 4.84933421539356018805823812814, 5.99912202698253601773987535907, 7.85250401913226081527003988553, 8.474076944337982711264776410486, 9.910611902728387012086470740501, 11.14713281637946271773117000662, 11.93542752930950848670594718744, 13.46608028965734558368467201644
