| L(s) = 1 | + 9·3-s + 107.·5-s + 149.·7-s + 81·9-s − 434.·11-s − 392.·13-s + 963.·15-s + 803.·17-s + 854.·19-s + 1.34e3·21-s − 4.59e3·23-s + 8.34e3·25-s + 729·27-s + 6.79e3·29-s − 4.79e3·31-s − 3.91e3·33-s + 1.59e4·35-s − 909.·37-s − 3.53e3·39-s − 2.37e3·41-s − 8.66e3·43-s + 8.67e3·45-s + 1.68e4·47-s + 5.41e3·49-s + 7.23e3·51-s + 1.08e4·53-s − 4.65e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.91·5-s + 1.14·7-s + 0.333·9-s − 1.08·11-s − 0.644·13-s + 1.10·15-s + 0.674·17-s + 0.543·19-s + 0.663·21-s − 1.81·23-s + 2.66·25-s + 0.192·27-s + 1.50·29-s − 0.896·31-s − 0.625·33-s + 2.20·35-s − 0.109·37-s − 0.371·39-s − 0.220·41-s − 0.714·43-s + 0.638·45-s + 1.11·47-s + 0.322·49-s + 0.389·51-s + 0.529·53-s − 2.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.207276984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.207276984\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| good | 5 | \( 1 - 107.T + 3.12e3T^{2} \) |
| 7 | \( 1 - 149.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 434.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 392.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 803.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 854.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.59e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.79e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.79e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 909.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.37e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.66e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.68e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.08e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 8.30e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.59e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.50e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.78e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.80e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.87e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.48e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.46e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.98e4T + 8.58e9T^{2} \) |
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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
−13.31474362258981328664402008445, −12.10912481332933460893494155649, −10.42098276578784588577857983225, −9.913199136562547562677378010591, −8.626802183863672773537982939927, −7.50694669914831601736673824687, −5.84140299252974182624626090693, −4.90464331602273466484279221205, −2.61657655144675772129530448179, −1.60951289621691539682008742832,
1.60951289621691539682008742832, 2.61657655144675772129530448179, 4.90464331602273466484279221205, 5.84140299252974182624626090693, 7.50694669914831601736673824687, 8.626802183863672773537982939927, 9.913199136562547562677378010591, 10.42098276578784588577857983225, 12.10912481332933460893494155649, 13.31474362258981328664402008445
