| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s + 64.7·5-s − 36·6-s − 91.1·7-s − 64·8-s + 81·9-s − 259.·10-s + 642.·11-s + 144·12-s + 597.·13-s + 364.·14-s + 582.·15-s + 256·16-s − 847.·17-s − 324·18-s − 1.41e3·19-s + 1.03e3·20-s − 820.·21-s − 2.56e3·22-s − 529·23-s − 576·24-s + 1.06e3·25-s − 2.38e3·26-s + 729·27-s − 1.45e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.15·5-s − 0.408·6-s − 0.702·7-s − 0.353·8-s + 0.333·9-s − 0.819·10-s + 1.60·11-s + 0.288·12-s + 0.979·13-s + 0.496·14-s + 0.668·15-s + 0.250·16-s − 0.710·17-s − 0.235·18-s − 0.899·19-s + 0.579·20-s − 0.405·21-s − 1.13·22-s − 0.208·23-s − 0.204·24-s + 0.342·25-s − 0.692·26-s + 0.192·27-s − 0.351·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.172028884\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.172028884\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 - 9T \) |
| 23 | \( 1 + 529T \) |
| good | 5 | \( 1 - 64.7T + 3.12e3T^{2} \) |
| 7 | \( 1 + 91.1T + 1.68e4T^{2} \) |
| 11 | \( 1 - 642.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 597.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 847.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.41e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 3.21e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.26e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.16e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.04e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.32e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.89e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 9.79e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.35e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.98e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.82e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.90e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.60e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.60e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.17e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.53e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.16e5T + 8.58e9T^{2} \) |
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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
−12.29366546587222275177250504746, −10.97878477418482270544473729227, −9.876186326255864382487220177377, −9.202881108280875760002813643483, −8.403848806688357486963549922066, −6.64609102970219500835240429009, −6.18588843549231633234251998356, −4.00712466853445889085580430097, −2.44348626834184429299673535471, −1.17149662571756724340817222654,
1.17149662571756724340817222654, 2.44348626834184429299673535471, 4.00712466853445889085580430097, 6.18588843549231633234251998356, 6.64609102970219500835240429009, 8.403848806688357486963549922066, 9.202881108280875760002813643483, 9.876186326255864382487220177377, 10.97878477418482270544473729227, 12.29366546587222275177250504746
