L(s) = 1 | + (0.959 − 0.281i)2-s + (−0.654 − 0.755i)3-s + (0.841 − 0.540i)4-s + (0.239 + 1.66i)5-s + (−0.841 − 0.540i)6-s + (0.415 − 0.909i)7-s + (0.654 − 0.755i)8-s + (−0.142 + 0.989i)9-s + (0.698 + 1.53i)10-s + (0.797 + 0.234i)11-s + (−0.959 − 0.281i)12-s + (0.142 − 0.989i)14-s + (1.10 − 1.27i)15-s + (0.415 − 0.909i)16-s + (1.10 + 0.708i)17-s + (0.142 + 0.989i)18-s + ⋯ |
L(s) = 1 | + (0.959 − 0.281i)2-s + (−0.654 − 0.755i)3-s + (0.841 − 0.540i)4-s + (0.239 + 1.66i)5-s + (−0.841 − 0.540i)6-s + (0.415 − 0.909i)7-s + (0.654 − 0.755i)8-s + (−0.142 + 0.989i)9-s + (0.698 + 1.53i)10-s + (0.797 + 0.234i)11-s + (−0.959 − 0.281i)12-s + (0.142 − 0.989i)14-s + (1.10 − 1.27i)15-s + (0.415 − 0.909i)16-s + (1.10 + 0.708i)17-s + (0.142 + 0.989i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.907089316\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.907089316\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.959 + 0.281i)T \) |
| 3 | \( 1 + (0.654 + 0.755i)T \) |
| 7 | \( 1 + (-0.415 + 0.909i)T \) |
| 23 | \( 1 + (0.841 - 0.540i)T \) |
good | 5 | \( 1 + (-0.239 - 1.66i)T + (-0.959 + 0.281i)T^{2} \) |
| 11 | \( 1 + (-0.797 - 0.234i)T + (0.841 + 0.540i)T^{2} \) |
| 13 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 17 | \( 1 + (-1.10 - 0.708i)T + (0.415 + 0.909i)T^{2} \) |
| 19 | \( 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2} \) |
| 29 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 31 | \( 1 + (-0.857 + 0.989i)T + (-0.142 - 0.989i)T^{2} \) |
| 37 | \( 1 + (-0.0405 + 0.281i)T + (-0.959 - 0.281i)T^{2} \) |
| 41 | \( 1 + (0.273 + 1.89i)T + (-0.959 + 0.281i)T^{2} \) |
| 43 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 59 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 61 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 67 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 71 | \( 1 + (0.273 - 0.0801i)T + (0.841 - 0.540i)T^{2} \) |
| 73 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 79 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 83 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 89 | \( 1 + (0.186 + 0.215i)T + (-0.142 + 0.989i)T^{2} \) |
| 97 | \( 1 + (0.959 - 0.281i)T^{2} \) |
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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
−9.907865115911801432864064820663, −8.052665830917224329523394772266, −7.46730393613292940869170511464, −6.71322568560418844882811701681, −6.21225488923472695847625637030, −5.58481783769455011886705234683, −4.16214117635469790252263436893, −3.66413040418208509025172109548, −2.30910442008975315843911995659, −1.57504565540675611673038274788,
1.38227375935753546782641795850, 2.80520196742009020400495003501, 4.11376830084301848218018618979, 4.74371206571795234893197184733, 5.18434034790591694811017190699, 6.04014517931157905381266961603, 6.58805895223468205295682713823, 8.113392018877384029919952337410, 8.630363040735180602095975897093, 9.312864361424629027740346245370