| L(s) = 1 | + 9·3-s + 25·5-s + 49·7-s − 162·9-s + 187·11-s + 627·13-s + 225·15-s + 1.81e3·17-s − 258·19-s + 441·21-s − 2.97e3·23-s + 625·25-s − 3.64e3·27-s + 1.29e3·29-s − 1.91e3·31-s + 1.68e3·33-s + 1.22e3·35-s + 6.57e3·37-s + 5.64e3·39-s + 6.67e3·41-s − 3.17e3·43-s − 4.05e3·45-s + 2.20e4·47-s + 2.40e3·49-s + 1.63e4·51-s + 2.61e4·53-s + 4.67e3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.465·11-s + 1.02·13-s + 0.258·15-s + 1.52·17-s − 0.163·19-s + 0.218·21-s − 1.17·23-s + 1/5·25-s − 0.962·27-s + 0.286·29-s − 0.358·31-s + 0.269·33-s + 0.169·35-s + 0.789·37-s + 0.594·39-s + 0.620·41-s − 0.262·43-s − 0.298·45-s + 1.45·47-s + 1/7·49-s + 0.878·51-s + 1.27·53-s + 0.208·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.358982142\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.358982142\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 - p^{2} T + p^{5} T^{2} \) |
| 11 | \( 1 - 17 p T + p^{5} T^{2} \) |
| 13 | \( 1 - 627 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1813 T + p^{5} T^{2} \) |
| 19 | \( 1 + 258 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2970 T + p^{5} T^{2} \) |
| 29 | \( 1 - 1299 T + p^{5} T^{2} \) |
| 31 | \( 1 + 1916 T + p^{5} T^{2} \) |
| 37 | \( 1 - 6578 T + p^{5} T^{2} \) |
| 41 | \( 1 - 6676 T + p^{5} T^{2} \) |
| 43 | \( 1 + 3178 T + p^{5} T^{2} \) |
| 47 | \( 1 - 22001 T + p^{5} T^{2} \) |
| 53 | \( 1 - 26168 T + p^{5} T^{2} \) |
| 59 | \( 1 + 3932 T + p^{5} T^{2} \) |
| 61 | \( 1 + 48740 T + p^{5} T^{2} \) |
| 67 | \( 1 - 44832 T + p^{5} T^{2} \) |
| 71 | \( 1 + 63736 T + p^{5} T^{2} \) |
| 73 | \( 1 - 60470 T + p^{5} T^{2} \) |
| 79 | \( 1 - 43721 T + p^{5} T^{2} \) |
| 83 | \( 1 + 1172 p T + p^{5} T^{2} \) |
| 89 | \( 1 - 45560 T + p^{5} T^{2} \) |
| 97 | \( 1 + 57295 T + p^{5} T^{2} \) |
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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
−9.886916939345642234567082327582, −9.016194647470853261834080638020, −8.290005276079899110812926417925, −7.51147412137894366301751430770, −6.12609671950279198989055381146, −5.59044256929898390279947709388, −4.13282764496662409314805535953, −3.20169637515775216717433336938, −2.04104787088836360183899849376, −0.913278798818510425948704305401,
0.913278798818510425948704305401, 2.04104787088836360183899849376, 3.20169637515775216717433336938, 4.13282764496662409314805535953, 5.59044256929898390279947709388, 6.12609671950279198989055381146, 7.51147412137894366301751430770, 8.290005276079899110812926417925, 9.016194647470853261834080638020, 9.886916939345642234567082327582
