| L(s) = 1 | + 17.5·3-s + 87.4·5-s + 76.9·7-s + 64.9·9-s − 99.2·11-s − 79.5·13-s + 1.53e3·15-s − 465.·17-s − 361·19-s + 1.34e3·21-s + 2.23e3·23-s + 4.52e3·25-s − 3.12e3·27-s + 757.·29-s + 2.71e3·31-s − 1.74e3·33-s + 6.72e3·35-s + 1.03e4·37-s − 1.39e3·39-s − 6.94e3·41-s − 2.10e3·43-s + 5.68e3·45-s − 2.05e4·47-s − 1.08e4·49-s − 8.17e3·51-s − 1.75e4·53-s − 8.67e3·55-s + ⋯ |
| L(s) = 1 | + 1.12·3-s + 1.56·5-s + 0.593·7-s + 0.267·9-s − 0.247·11-s − 0.130·13-s + 1.76·15-s − 0.391·17-s − 0.229·19-s + 0.668·21-s + 0.879·23-s + 1.44·25-s − 0.824·27-s + 0.167·29-s + 0.507·31-s − 0.278·33-s + 0.928·35-s + 1.23·37-s − 0.146·39-s − 0.645·41-s − 0.173·43-s + 0.418·45-s − 1.35·47-s − 0.647·49-s − 0.440·51-s − 0.859·53-s − 0.386·55-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
| \(L(3)\) |
\(\approx\) |
\(3.232719759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.232719759\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 17.5T + 243T^{2} \) |
| 5 | \( 1 - 87.4T + 3.12e3T^{2} \) |
| 7 | \( 1 - 76.9T + 1.68e4T^{2} \) |
| 11 | \( 1 + 99.2T + 1.61e5T^{2} \) |
| 13 | \( 1 + 79.5T + 3.71e5T^{2} \) |
| 17 | \( 1 + 465.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 2.23e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 757.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.71e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.03e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.94e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.10e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.05e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.75e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.45e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.09e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.82e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.88e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.36e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.90e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.94e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.89e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.74e4T + 8.58e9T^{2} \) |
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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
−13.67976121231502930489182554393, −12.89518923131157052972504820759, −11.14761601619831445480702423221, −9.880041180394734203892525922465, −9.039202755430444181666244516576, −7.972373649925004474354322312758, −6.35828208620470384198163370998, −4.93379466306303632770095227429, −2.87019025177418279132846485714, −1.74145827011482975363307579674,
1.74145827011482975363307579674, 2.87019025177418279132846485714, 4.93379466306303632770095227429, 6.35828208620470384198163370998, 7.972373649925004474354322312758, 9.039202755430444181666244516576, 9.880041180394734203892525922465, 11.14761601619831445480702423221, 12.89518923131157052972504820759, 13.67976121231502930489182554393
