| L(s) = 1 | − 8.01·3-s + 93.2·5-s + 174.·7-s − 178.·9-s − 10.6·11-s + 305.·13-s − 747.·15-s − 605.·17-s − 984.·19-s − 1.40e3·21-s + 4.48e3·23-s + 5.57e3·25-s + 3.38e3·27-s + 3.26e3·29-s − 961·31-s + 85.3·33-s + 1.62e4·35-s + 2.57e3·37-s − 2.44e3·39-s − 1.25e4·41-s + 1.54e4·43-s − 1.66e4·45-s + 2.36e4·47-s + 1.37e4·49-s + 4.85e3·51-s − 2.36e4·53-s − 992.·55-s + ⋯ |
| L(s) = 1 | − 0.514·3-s + 1.66·5-s + 1.34·7-s − 0.735·9-s − 0.0265·11-s + 0.500·13-s − 0.858·15-s − 0.508·17-s − 0.625·19-s − 0.693·21-s + 1.76·23-s + 1.78·25-s + 0.892·27-s + 0.720·29-s − 0.179·31-s + 0.0136·33-s + 2.24·35-s + 0.308·37-s − 0.257·39-s − 1.16·41-s + 1.27·43-s − 1.22·45-s + 1.55·47-s + 0.817·49-s + 0.261·51-s − 1.15·53-s − 0.0442·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.394785148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.394785148\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + 961T \) |
| good | 3 | \( 1 + 8.01T + 243T^{2} \) |
| 5 | \( 1 - 93.2T + 3.12e3T^{2} \) |
| 7 | \( 1 - 174.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 10.6T + 1.61e5T^{2} \) |
| 13 | \( 1 - 305.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 605.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 984.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.48e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.26e3T + 2.05e7T^{2} \) |
| 37 | \( 1 - 2.57e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.25e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.54e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.36e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.36e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 7.43e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.12e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.19e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.31e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.97e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.50e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.03e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.11e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.52e3T + 8.58e9T^{2} \) |
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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
−12.51869114142656049493146614062, −11.11498634992078147855690206281, −10.71850873745352957896624432057, −9.249876780388782291553653367993, −8.400063400453439165863457959387, −6.68382848207408357091728816174, −5.65449089839642605710919211951, −4.82383717444235248483058881595, −2.50135896657747747484831887840, −1.21778139780511622250447486670,
1.21778139780511622250447486670, 2.50135896657747747484831887840, 4.82383717444235248483058881595, 5.65449089839642605710919211951, 6.68382848207408357091728816174, 8.400063400453439165863457959387, 9.249876780388782291553653367993, 10.71850873745352957896624432057, 11.11498634992078147855690206281, 12.51869114142656049493146614062
