L(s) = 1 | + (−0.936 − 0.349i)3-s + (−0.989 + 0.142i)5-s + (−0.800 − 0.599i)7-s + (0.755 + 0.654i)9-s + (0.142 − 0.989i)11-s + (−0.349 + 0.936i)13-s + (0.977 + 0.212i)15-s + (−0.540 + 0.841i)17-s + (−0.997 + 0.0713i)19-s + (0.540 + 0.841i)21-s + (0.0713 + 0.997i)23-s + (0.959 − 0.281i)25-s + (−0.479 − 0.877i)27-s + (−0.599 + 0.800i)29-s + (0.0713 − 0.997i)31-s + ⋯ |
L(s) = 1 | + (−0.936 − 0.349i)3-s + (−0.989 + 0.142i)5-s + (−0.800 − 0.599i)7-s + (0.755 + 0.654i)9-s + (0.142 − 0.989i)11-s + (−0.349 + 0.936i)13-s + (0.977 + 0.212i)15-s + (−0.540 + 0.841i)17-s + (−0.997 + 0.0713i)19-s + (0.540 + 0.841i)21-s + (0.0713 + 0.997i)23-s + (0.959 − 0.281i)25-s + (−0.479 − 0.877i)27-s + (−0.599 + 0.800i)29-s + (0.0713 − 0.997i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5035222209 - 0.07788832377i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5035222209 - 0.07788832377i\) |
\(L(1)\) |
\(\approx\) |
\(0.5543492862 - 0.06141859553i\) |
\(L(1)\) |
\(\approx\) |
\(0.5543492862 - 0.06141859553i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 89 | \( 1 \) |
good | 3 | \( 1 + (-0.936 - 0.349i)T \) |
| 5 | \( 1 + (-0.989 + 0.142i)T \) |
| 7 | \( 1 + (-0.800 - 0.599i)T \) |
| 11 | \( 1 + (0.142 - 0.989i)T \) |
| 13 | \( 1 + (-0.349 + 0.936i)T \) |
| 17 | \( 1 + (-0.540 + 0.841i)T \) |
| 19 | \( 1 + (-0.997 + 0.0713i)T \) |
| 23 | \( 1 + (0.0713 + 0.997i)T \) |
| 29 | \( 1 + (-0.599 + 0.800i)T \) |
| 31 | \( 1 + (0.0713 - 0.997i)T \) |
| 37 | \( 1 + (-0.707 - 0.707i)T \) |
| 41 | \( 1 + (-0.349 - 0.936i)T \) |
| 43 | \( 1 + (0.599 + 0.800i)T \) |
| 47 | \( 1 + (-0.909 - 0.415i)T \) |
| 53 | \( 1 + (0.909 - 0.415i)T \) |
| 59 | \( 1 + (0.936 - 0.349i)T \) |
| 61 | \( 1 + (0.877 - 0.479i)T \) |
| 67 | \( 1 + (0.415 + 0.909i)T \) |
| 71 | \( 1 + (0.989 + 0.142i)T \) |
| 73 | \( 1 + (0.654 + 0.755i)T \) |
| 79 | \( 1 + (0.755 - 0.654i)T \) |
| 83 | \( 1 + (0.977 - 0.212i)T \) |
| 97 | \( 1 + (-0.142 - 0.989i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−22.705799527272476016728268041904, −22.19809571788940730452294976638, −21.005847680451595123567661708336, −20.21187064988965756002657068709, −19.394356921143087109662981546011, −18.50916514932065814848447290085, −17.72677239391477713463233175506, −16.81322815497607113046593778803, −16.06338085293763764083939724092, −15.28064206330348099775248715541, −14.947390028979382579247364508148, −13.17536541640888362432667775735, −12.400638009170586716685584076296, −12.024097813759443384732498770018, −10.964167268353421554764862670852, −10.15400604076058914027977723029, −9.29313949297835467636383667461, −8.27086797376186046131929567445, −7.05145732546949107608539797833, −6.49869924035990744384292971957, −5.231650197272031667307320196012, −4.56729161546574080248974967513, −3.568769480706251558180152736686, −2.38146790772751517777343431803, −0.55534230298386248850417630892,
0.563530670020375040651193630746, 2.009860666889017533711042753904, 3.64549557037954193975484457204, 4.130943120552055583462857316734, 5.42273827370943554925079956492, 6.51055690840340966549378632325, 6.9790818247830629516462565346, 7.983571436256987564488095992890, 9.03396342879426959475875179715, 10.24593923390021970579128655472, 11.08058125549391006814886882971, 11.55905856535999580525679166014, 12.63384425859900609971644611020, 13.19688201625498999700195994654, 14.27556374921421258465392900218, 15.38848760878804840636755360706, 16.23159662308454956982728002337, 16.74130719249644976481929957040, 17.4927103176021835157702567085, 18.76885489998998975424269807542, 19.25223925410752032239835157890, 19.71521007718419029929060069145, 21.1272147567300331061945906550, 21.98958660331550349043979159985, 22.583789555860862641226394196