L(s) = 1 | + (0.415 + 0.909i)2-s + (0.959 − 0.281i)3-s + (−0.654 + 0.755i)4-s + (−0.841 − 0.540i)5-s + (0.654 + 0.755i)6-s + (−0.959 − 0.281i)8-s + (0.841 − 0.540i)9-s + (0.142 − 0.989i)10-s + (−0.698 + 1.53i)11-s + (−0.415 + 0.909i)12-s + (−0.0405 + 0.281i)13-s + (−0.959 − 0.281i)15-s + (−0.142 − 0.989i)16-s + (1.25 + 1.45i)17-s + (0.841 + 0.540i)18-s + ⋯ |
L(s) = 1 | + (0.415 + 0.909i)2-s + (0.959 − 0.281i)3-s + (−0.654 + 0.755i)4-s + (−0.841 − 0.540i)5-s + (0.654 + 0.755i)6-s + (−0.959 − 0.281i)8-s + (0.841 − 0.540i)9-s + (0.142 − 0.989i)10-s + (−0.698 + 1.53i)11-s + (−0.415 + 0.909i)12-s + (−0.0405 + 0.281i)13-s + (−0.959 − 0.281i)15-s + (−0.142 − 0.989i)16-s + (1.25 + 1.45i)17-s + (0.841 + 0.540i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.633428234\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.633428234\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.415 - 0.909i)T \) |
| 3 | \( 1 + (-0.959 + 0.281i)T \) |
| 5 | \( 1 + (0.841 + 0.540i)T \) |
| 23 | \( 1 + (-0.841 + 0.540i)T \) |
good | 7 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 11 | \( 1 + (0.698 - 1.53i)T + (-0.654 - 0.755i)T^{2} \) |
| 13 | \( 1 + (0.0405 - 0.281i)T + (-0.959 - 0.281i)T^{2} \) |
| 17 | \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \) |
| 19 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 29 | \( 1 + (-0.544 - 0.627i)T + (-0.142 + 0.989i)T^{2} \) |
| 31 | \( 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2} \) |
| 37 | \( 1 + (0.698 - 0.449i)T + (0.415 - 0.909i)T^{2} \) |
| 41 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 43 | \( 1 + (0.273 - 0.0801i)T + (0.841 - 0.540i)T^{2} \) |
| 47 | \( 1 + 1.91T + T^{2} \) |
| 53 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 59 | \( 1 + (-0.239 + 1.66i)T + (-0.959 - 0.281i)T^{2} \) |
| 61 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 67 | \( 1 + (-0.544 - 1.19i)T + (-0.654 + 0.755i)T^{2} \) |
| 71 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 73 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 79 | \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \) |
| 83 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 89 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 97 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
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
−8.805772590853862859602677318806, −8.179509224886073137207013119336, −7.84856546480728615937165490594, −7.01331693657014054727751719864, −6.45270065845748855113339549741, −5.01305575974127110900272216348, −4.66440370495859245944890399126, −3.68560707281682792870305639230, −2.98038540937831050189776323749, −1.53667912780565316277099805157,
0.896407822977414478026355187908, 2.59262602469358292486189879179, 3.14906072370255789724391574239, 3.56337393188397553700033846157, 4.73013288676027134893267666253, 5.34924829137564747547608021649, 6.48513355919234652016550001636, 7.57655827483641778582077307918, 8.160430025457659897055258267744, 8.783901532296663289037235615138