| L(s) = 1 | + 2.00·2-s − 26.2·3-s − 27.9·4-s + 74.6·5-s − 52.6·6-s + 126.·7-s − 120.·8-s + 445.·9-s + 149.·10-s + 732.·11-s + 733.·12-s − 948.·13-s + 253.·14-s − 1.95e3·15-s + 653.·16-s + 447.·17-s + 894.·18-s + 506.·19-s − 2.08e3·20-s − 3.31e3·21-s + 1.47e3·22-s + 3.64e3·23-s + 3.16e3·24-s + 2.44e3·25-s − 1.90e3·26-s − 5.31e3·27-s − 3.53e3·28-s + ⋯ |
| L(s) = 1 | + 0.355·2-s − 1.68·3-s − 0.873·4-s + 1.33·5-s − 0.597·6-s + 0.974·7-s − 0.665·8-s + 1.83·9-s + 0.474·10-s + 1.82·11-s + 1.47·12-s − 1.55·13-s + 0.345·14-s − 2.24·15-s + 0.637·16-s + 0.375·17-s + 0.650·18-s + 0.322·19-s − 1.16·20-s − 1.63·21-s + 0.648·22-s + 1.43·23-s + 1.11·24-s + 0.783·25-s − 0.552·26-s − 1.40·27-s − 0.851·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.280127727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.280127727\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 41 | \( 1 - 1.68e3T \) |
| good | 2 | \( 1 - 2.00T + 32T^{2} \) |
| 3 | \( 1 + 26.2T + 243T^{2} \) |
| 5 | \( 1 - 74.6T + 3.12e3T^{2} \) |
| 7 | \( 1 - 126.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 732.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 948.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 447.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 506.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.64e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.03e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 528.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.02e4T + 6.93e7T^{2} \) |
| 43 | \( 1 + 3.66e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.67e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 6.95e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.21e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.83e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.17e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.52e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.13e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.16e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.23e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 444.T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.53e4T + 8.58e9T^{2} \) |
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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
−14.72479580404255849720948113740, −13.95459392019493581997541106150, −12.48532788449071637807309015694, −11.71974566914130275247866313340, −10.22528185595576635726277977091, −9.202656520981390101566071648790, −6.75816000656926975003193276149, −5.46113623922644393671468000560, −4.69737856711548811543386406946, −1.16133380844912504594622500734,
1.16133380844912504594622500734, 4.69737856711548811543386406946, 5.46113623922644393671468000560, 6.75816000656926975003193276149, 9.202656520981390101566071648790, 10.22528185595576635726277977091, 11.71974566914130275247866313340, 12.48532788449071637807309015694, 13.95459392019493581997541106150, 14.72479580404255849720948113740
