| L(s) = 1 | − 8·2-s + 10.9·3-s + 64·4-s + 82.9·5-s − 87.5·6-s − 343·7-s − 512·8-s − 2.06e3·9-s − 663.·10-s − 1.08e3·11-s + 700.·12-s − 2.19e3·13-s + 2.74e3·14-s + 907.·15-s + 4.09e3·16-s − 2.84e4·17-s + 1.65e4·18-s + 4.89e4·19-s + 5.30e3·20-s − 3.75e3·21-s + 8.71e3·22-s + 5.73e4·23-s − 5.60e3·24-s − 7.12e4·25-s + 1.75e4·26-s − 4.65e4·27-s − 2.19e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.233·3-s + 0.5·4-s + 0.296·5-s − 0.165·6-s − 0.377·7-s − 0.353·8-s − 0.945·9-s − 0.209·10-s − 0.246·11-s + 0.116·12-s − 0.277·13-s + 0.267·14-s + 0.0694·15-s + 0.250·16-s − 1.40·17-s + 0.668·18-s + 1.63·19-s + 0.148·20-s − 0.0884·21-s + 0.174·22-s + 0.983·23-s − 0.0827·24-s − 0.912·25-s + 0.196·26-s − 0.455·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.222912145\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.222912145\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 7 | \( 1 + 343T \) |
| 13 | \( 1 + 2.19e3T \) |
| good | 3 | \( 1 - 10.9T + 2.18e3T^{2} \) |
| 5 | \( 1 - 82.9T + 7.81e4T^{2} \) |
| 11 | \( 1 + 1.08e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 2.84e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.89e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.73e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.41e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.05e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 8.72e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 7.88e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.95e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.43e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 5.36e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.21e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.04e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.79e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.14e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.61e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.04e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.13e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 8.08e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.32e7T + 8.07e13T^{2} \) |
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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
−11.26797392154325512685136538072, −10.18380314821059751738022015683, −9.267055928223956064322119256844, −8.491579894680586141102123520567, −7.33560048200861937219217248305, −6.27109244323100305737588058530, −5.07362868670274311636466342307, −3.24471071490612304181758284026, −2.25087642962469931783479832529, −0.63990171195041374276192027935,
0.63990171195041374276192027935, 2.25087642962469931783479832529, 3.24471071490612304181758284026, 5.07362868670274311636466342307, 6.27109244323100305737588058530, 7.33560048200861937219217248305, 8.491579894680586141102123520567, 9.267055928223956064322119256844, 10.18380314821059751738022015683, 11.26797392154325512685136538072
