| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s − 1.62·5-s + 6·6-s + 7·7-s + 8·8-s + 9·9-s − 3.24·10-s + 64.4·11-s + 12·12-s − 71.6·13-s + 14·14-s − 4.86·15-s + 16·16-s + 105.·17-s + 18·18-s − 32.6·19-s − 6.48·20-s + 21·21-s + 128.·22-s + 23·23-s + 24·24-s − 122.·25-s − 143.·26-s + 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.145·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.102·10-s + 1.76·11-s + 0.288·12-s − 1.52·13-s + 0.267·14-s − 0.0837·15-s + 0.250·16-s + 1.50·17-s + 0.235·18-s − 0.394·19-s − 0.0725·20-s + 0.218·21-s + 1.24·22-s + 0.208·23-s + 0.204·24-s − 0.978·25-s − 1.08·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.769995730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.769995730\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| 23 | \( 1 - 23T \) |
| good | 5 | \( 1 + 1.62T + 125T^{2} \) |
| 11 | \( 1 - 64.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 71.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 105.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 32.6T + 6.85e3T^{2} \) |
| 29 | \( 1 - 268.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 66.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 294.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 298.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 168.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 455.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 540.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 406.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 658.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 685.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 2.75T + 3.57e5T^{2} \) |
| 73 | \( 1 - 400.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 767.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 987.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 689.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.15e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.712690667832034593749599329189, −8.775140750426666696891337980377, −7.82351454580745088542203924625, −7.10461531141111928179155845840, −6.22962872798957619135548751056, −5.08726534679606770051726646297, −4.24202910003682265311604088421, −3.39483253635467080064777367758, −2.27947616098872259289676308286, −1.11403954895499479363209576639,
1.11403954895499479363209576639, 2.27947616098872259289676308286, 3.39483253635467080064777367758, 4.24202910003682265311604088421, 5.08726534679606770051726646297, 6.22962872798957619135548751056, 7.10461531141111928179155845840, 7.82351454580745088542203924625, 8.775140750426666696891337980377, 9.712690667832034593749599329189
