L(s) = 1 | + (−0.826 + 0.563i)2-s + (0.997 − 0.925i)3-s + (0.365 − 0.930i)4-s + (−0.302 + 1.32i)6-s + (−0.974 + 0.222i)7-s + (0.222 + 0.974i)8-s + (0.0635 − 0.848i)9-s + (0.0841 + 1.12i)11-s + (−0.496 − 1.26i)12-s + (1.48 + 0.716i)13-s + (0.680 − 0.733i)14-s + (−0.733 − 0.680i)16-s + (−0.988 − 0.149i)17-s + (0.425 + 0.736i)18-s + (−0.766 + 1.12i)21-s + (−0.702 − 0.880i)22-s + ⋯ |
L(s) = 1 | + (−0.826 + 0.563i)2-s + (0.997 − 0.925i)3-s + (0.365 − 0.930i)4-s + (−0.302 + 1.32i)6-s + (−0.974 + 0.222i)7-s + (0.222 + 0.974i)8-s + (0.0635 − 0.848i)9-s + (0.0841 + 1.12i)11-s + (−0.496 − 1.26i)12-s + (1.48 + 0.716i)13-s + (0.680 − 0.733i)14-s + (−0.733 − 0.680i)16-s + (−0.988 − 0.149i)17-s + (0.425 + 0.736i)18-s + (−0.766 + 1.12i)21-s + (−0.702 − 0.880i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.084405823\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.084405823\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.826 - 0.563i)T \) |
| 7 | \( 1 + (0.974 - 0.222i)T \) |
| 17 | \( 1 + (0.988 + 0.149i)T \) |
good | 3 | \( 1 + (-0.997 + 0.925i)T + (0.0747 - 0.997i)T^{2} \) |
| 5 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 11 | \( 1 + (-0.0841 - 1.12i)T + (-0.988 + 0.149i)T^{2} \) |
| 13 | \( 1 + (-1.48 - 0.716i)T + (0.623 + 0.781i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.858 - 0.129i)T + (0.955 - 0.294i)T^{2} \) |
| 29 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 31 | \( 1 + (-0.974 - 1.68i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 41 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 43 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 47 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 53 | \( 1 + (-0.698 + 1.77i)T + (-0.733 - 0.680i)T^{2} \) |
| 59 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 61 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.367 - 0.460i)T + (-0.222 + 0.974i)T^{2} \) |
| 73 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 79 | \( 1 + (0.997 - 1.72i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 89 | \( 1 + (-0.109 + 1.46i)T + (-0.988 - 0.149i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.601775688337283700608350696991, −8.430785047807776942931765474445, −7.30313566963934862664605635849, −6.72050677911481539877920795359, −6.50769190276427528181751078812, −5.32651785516326515500264831298, −4.16858002937260766617424397861, −3.03328810493551691038282577644, −2.11194521950529074064507313042, −1.35483410091838821716398889191,
0.812594350950895735525040373257, 2.47916111899799841675566680203, 3.09778967226509103587018160304, 3.81269549821643372520611551279, 4.30654694241900634250566130921, 6.04786339315557409410555776879, 6.39877422783154697418861513609, 7.65554606257011964193984379702, 8.421067427998082022721063648268, 8.742294386623724572680217404746