| L(s) = 1 | − 1.93i·2-s − 1.73·4-s − 3.31i·5-s + (2.34 + 1.22i)7-s − 0.517i·8-s + 3·9-s − 6.40·10-s + 6.40·13-s + (2.36 − 4.53i)14-s − 4.46·16-s + 1.71·17-s − 5.79i·18-s + 4.69·19-s + 5.74i·20-s − 4.73·23-s + ⋯ |
| L(s) = 1 | − 1.36i·2-s − 0.866·4-s − 1.48i·5-s + (0.886 + 0.462i)7-s − 0.183i·8-s + 9-s − 2.02·10-s + 1.77·13-s + (0.632 − 1.21i)14-s − 1.11·16-s + 0.416·17-s − 1.36i·18-s + 1.07·19-s + 1.28i·20-s − 0.986·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 + 0.514i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.534967 - 1.93277i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.534967 - 1.93277i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.34 - 1.22i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.93iT - 2T^{2} \) |
| 3 | \( 1 - 3T^{2} \) |
| 5 | \( 1 + 3.31iT - 5T^{2} \) |
| 13 | \( 1 - 6.40T + 13T^{2} \) |
| 17 | \( 1 - 1.71T + 17T^{2} \) |
| 19 | \( 1 - 4.69T + 19T^{2} \) |
| 23 | \( 1 + 4.73T + 23T^{2} \) |
| 29 | \( 1 - 3.72iT - 29T^{2} \) |
| 31 | \( 1 - 2.42iT - 31T^{2} \) |
| 37 | \( 1 + 1.53T + 37T^{2} \) |
| 41 | \( 1 + 6.40T + 41T^{2} \) |
| 43 | \( 1 - 8.76iT - 43T^{2} \) |
| 47 | \( 1 + 2.42iT - 47T^{2} \) |
| 53 | \( 1 + 9.92T + 53T^{2} \) |
| 59 | \( 1 - 6.63iT - 59T^{2} \) |
| 61 | \( 1 + 8.12T + 61T^{2} \) |
| 67 | \( 1 + 9.66T + 67T^{2} \) |
| 71 | \( 1 + 0.535T + 71T^{2} \) |
| 73 | \( 1 + 4.69T + 73T^{2} \) |
| 79 | \( 1 + 0.656iT - 79T^{2} \) |
| 83 | \( 1 + 9.38T + 83T^{2} \) |
| 89 | \( 1 + 5.74iT - 89T^{2} \) |
| 97 | \( 1 + 14.8iT - 97T^{2} \) |
| 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
−9.945604295097002953549591889305, −9.091110680425308776650601750016, −8.517521459733711976793589687717, −7.57918892580768122074912332689, −6.10999457495263950942589180995, −5.01428378924992940008186839190, −4.29264334547320833663226244991, −3.31910391642682370795592371583, −1.50250688262833871268849553285, −1.32951711163343855221565604431,
1.72762786361179365377995926205, 3.40890822110023567833459507348, 4.37023207581143627835905092057, 5.63631635608858743070732676253, 6.37201255729525726930550516438, 7.14001223449378706488499753140, 7.72042979810892252603373855023, 8.403451640128728267648949035037, 9.682357383440821680158344650570, 10.57250751454439884691241734472
