| L(s) = 1 | + 17.7·2-s − 198.·4-s + 739.·5-s − 6.02e3·7-s − 1.25e4·8-s + 1.31e4·10-s − 1.84e4·11-s + 1.17e5·13-s − 1.06e5·14-s − 1.21e5·16-s + 4.68e5·17-s + 8.34e5·19-s − 1.46e5·20-s − 3.26e5·22-s − 1.32e6·23-s − 1.40e6·25-s + 2.08e6·26-s + 1.19e6·28-s − 6.42e4·29-s + 7.41e6·31-s + 4.28e6·32-s + 8.29e6·34-s − 4.45e6·35-s + 3.68e6·37-s + 1.47e7·38-s − 9.30e6·40-s + 2.53e7·41-s + ⋯ |
| L(s) = 1 | + 0.783·2-s − 0.386·4-s + 0.529·5-s − 0.948·7-s − 1.08·8-s + 0.414·10-s − 0.379·11-s + 1.14·13-s − 0.742·14-s − 0.463·16-s + 1.35·17-s + 1.46·19-s − 0.204·20-s − 0.297·22-s − 0.983·23-s − 0.719·25-s + 0.894·26-s + 0.367·28-s − 0.0168·29-s + 1.44·31-s + 0.722·32-s + 1.06·34-s − 0.502·35-s + 0.322·37-s + 1.15·38-s − 0.574·40-s + 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.553712446\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.553712446\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 17.7T + 512T^{2} \) |
| 5 | \( 1 - 739.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 6.02e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.84e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.17e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 4.68e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 8.34e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.32e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 6.42e4T + 1.45e13T^{2} \) |
| 31 | \( 1 - 7.41e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 3.68e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.53e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.42e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 7.03e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 3.10e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.53e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.55e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.39e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.45e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.01e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.59e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.05e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 8.60e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.48e7T + 7.60e17T^{2} \) |
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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
−12.72837643692272685892768646732, −11.75999822076681290300688058909, −10.07980136589748068554620439514, −9.383856988531253749802593532759, −7.936512360438357485376511808210, −6.17356906442839206302603917553, −5.52458648484936601648929499077, −3.91113305022698581968570456926, −2.90113392315615546691943121808, −0.851661001140223086632372079579,
0.851661001140223086632372079579, 2.90113392315615546691943121808, 3.91113305022698581968570456926, 5.52458648484936601648929499077, 6.17356906442839206302603917553, 7.936512360438357485376511808210, 9.383856988531253749802593532759, 10.07980136589748068554620439514, 11.75999822076681290300688058909, 12.72837643692272685892768646732
