L(s) = 1 | + 2-s + 4-s + (2.00 + 0.997i)5-s + 8-s + (2.00 + 0.997i)10-s − 0.311i·11-s + 1.62·13-s + 16-s − 1.60i·17-s − 4.54i·19-s + (2.00 + 0.997i)20-s − 0.311i·22-s + 4.42·23-s + (3.00 + 3.99i)25-s + 1.62·26-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + (0.894 + 0.446i)5-s + 0.353·8-s + (0.632 + 0.315i)10-s − 0.0939i·11-s + 0.451·13-s + 0.250·16-s − 0.390i·17-s − 1.04i·19-s + (0.447 + 0.223i)20-s − 0.0664i·22-s + 0.922·23-s + (0.601 + 0.798i)25-s + 0.319·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00419i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00419i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.983387744\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.983387744\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.00 - 0.997i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 0.311iT - 11T^{2} \) |
| 13 | \( 1 - 1.62T + 13T^{2} \) |
| 17 | \( 1 + 1.60iT - 17T^{2} \) |
| 19 | \( 1 + 4.54iT - 19T^{2} \) |
| 23 | \( 1 - 4.42T + 23T^{2} \) |
| 29 | \( 1 + 1.79iT - 29T^{2} \) |
| 31 | \( 1 - 0.415iT - 31T^{2} \) |
| 37 | \( 1 + 2.16iT - 37T^{2} \) |
| 41 | \( 1 - 3.39T + 41T^{2} \) |
| 43 | \( 1 + 0.812iT - 43T^{2} \) |
| 47 | \( 1 - 0.316iT - 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 + 9.94iT - 61T^{2} \) |
| 67 | \( 1 - 14.8iT - 67T^{2} \) |
| 71 | \( 1 + 2.38iT - 71T^{2} \) |
| 73 | \( 1 - 6.13T + 73T^{2} \) |
| 79 | \( 1 - 9.32T + 79T^{2} \) |
| 83 | \( 1 - 9.49iT - 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−8.364080285092405284118810354660, −7.31353090085512691993453077409, −6.82317014596712162862552418258, −6.10796155074027140031325759967, −5.39956438648487965562649091190, −4.75776094536981761658166924174, −3.74961199743422012067885314990, −2.86914875127445970106079198838, −2.23337785861896186328155130014, −1.01617112533107961536769184186,
1.13606103339452620806718984374, 1.97565280014328693000723147881, 2.96772059447736219953681277278, 3.86570445345509413813078614441, 4.70164515279918906973663321145, 5.43951184520175437774302992036, 6.04152130810725982757054182335, 6.65957420593055103690892358458, 7.56517506577595012966129573250, 8.372187293857472038496034046479