| 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 | \( 1 - 14.8iT - 97T^{2} \) |
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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
−10.57250751454439884691241734472, −9.682357383440821680158344650570, −8.403451640128728267648949035037, −7.72042979810892252603373855023, −7.14001223449378706488499753140, −6.37201255729525726930550516438, −5.63631635608858743070732676253, −4.37023207581143627835905092057, −3.40890822110023567833459507348, −1.72762786361179365377995926205,
1.32951711163343855221565604431, 1.50250688262833871268849553285, 3.31910391642682370795592371583, 4.29264334547320833663226244991, 5.01428378924992940008186839190, 6.10999457495263950942589180995, 7.57918892580768122074912332689, 8.517521459733711976793589687717, 9.091110680425308776650601750016, 9.945604295097002953549591889305
