L(s) = 1 | + (0.0475 − 0.998i)2-s + (−0.928 − 0.371i)3-s + (−0.995 − 0.0950i)4-s + (−0.235 − 0.971i)5-s + (−0.415 + 0.909i)6-s + (−0.142 + 0.989i)8-s + (0.723 + 0.690i)9-s + (−0.981 + 0.189i)10-s + (0.0475 + 0.998i)11-s + (0.888 + 0.458i)12-s + (0.654 + 0.755i)13-s + (−0.142 + 0.989i)15-s + (0.981 + 0.189i)16-s + (−0.580 − 0.814i)17-s + (0.723 − 0.690i)18-s + (−0.580 + 0.814i)19-s + ⋯ |
L(s) = 1 | + (0.0475 − 0.998i)2-s + (−0.928 − 0.371i)3-s + (−0.995 − 0.0950i)4-s + (−0.235 − 0.971i)5-s + (−0.415 + 0.909i)6-s + (−0.142 + 0.989i)8-s + (0.723 + 0.690i)9-s + (−0.981 + 0.189i)10-s + (0.0475 + 0.998i)11-s + (0.888 + 0.458i)12-s + (0.654 + 0.755i)13-s + (−0.142 + 0.989i)15-s + (0.981 + 0.189i)16-s + (−0.580 − 0.814i)17-s + (0.723 − 0.690i)18-s + (−0.580 + 0.814i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.683 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.683 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8280182191 - 0.3587401910i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8280182191 - 0.3587401910i\) |
\(L(1)\) |
\(\approx\) |
\(0.6116955632 - 0.3813146642i\) |
\(L(1)\) |
\(\approx\) |
\(0.6116955632 - 0.3813146642i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + (0.0475 - 0.998i)T \) |
| 3 | \( 1 + (-0.928 - 0.371i)T \) |
| 5 | \( 1 + (-0.235 - 0.971i)T \) |
| 11 | \( 1 + (0.0475 + 0.998i)T \) |
| 13 | \( 1 + (0.654 + 0.755i)T \) |
| 17 | \( 1 + (-0.580 - 0.814i)T \) |
| 19 | \( 1 + (-0.580 + 0.814i)T \) |
| 29 | \( 1 + (0.415 - 0.909i)T \) |
| 31 | \( 1 + (0.786 + 0.618i)T \) |
| 37 | \( 1 + (0.723 + 0.690i)T \) |
| 41 | \( 1 + (0.959 + 0.281i)T \) |
| 43 | \( 1 + (-0.142 - 0.989i)T \) |
| 47 | \( 1 + (0.5 + 0.866i)T \) |
| 53 | \( 1 + (-0.327 - 0.945i)T \) |
| 59 | \( 1 + (-0.981 + 0.189i)T \) |
| 61 | \( 1 + (-0.928 + 0.371i)T \) |
| 67 | \( 1 + (-0.888 + 0.458i)T \) |
| 71 | \( 1 + (0.841 + 0.540i)T \) |
| 73 | \( 1 + (0.995 + 0.0950i)T \) |
| 79 | \( 1 + (-0.327 + 0.945i)T \) |
| 83 | \( 1 + (0.959 - 0.281i)T \) |
| 89 | \( 1 + (0.786 - 0.618i)T \) |
| 97 | \( 1 + (0.959 + 0.281i)T \) |
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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
−27.505497383770431260896431876952, −26.605016803204485994658066171214, −25.97933671264638450360259361974, −24.614399874809124429631267774998, −23.6218032051431047759073195436, −22.9845387043204904166208591143, −22.00641130560288305422338447284, −21.462388886649943918682116538387, −19.48188686504644606776018237242, −18.373579546318955075775611094201, −17.70644355505526138844928114773, −16.67774617056534224470873571019, −15.63580801349092159273575892725, −15.069566710317690329140541270699, −13.768906312014329543473760064483, −12.659548654305080079662652051653, −11.12337375315968980477220521595, −10.48892440208042345997696281808, −9.01823497620988622207086739610, −7.76227839993655625296374849848, −6.42969236607937876459352102806, −5.96292700526280549297113526772, −4.484185867298433687268781045641, −3.342166409309727438920517609733, −0.51729939015016439121861678531,
0.94071450954699615654481158473, 2.05901606366629317169637720268, 4.21039677174653023052967928921, 4.84020145058971203546816402978, 6.24017671989116527656271715624, 7.843103925274306974546431269174, 9.11041664943295687557259311798, 10.18379091577663108511188120258, 11.43190360234638729994199887461, 12.11662184929315711874521726830, 12.9330926783311365046984009646, 13.89178981945016931001843988932, 15.6271556807201795845849896935, 16.79007622000318402605187179231, 17.62932113763071114163602246343, 18.59501535205131594948464034638, 19.58227144655218345859675332269, 20.66035663484648420494714852100, 21.392137949807540985069906385347, 22.71470286874205588925976487707, 23.25542386663563229605256505496, 24.20051186047557060561745288304, 25.36933744663426186133022900056, 26.98589225594906885100217941607, 27.74669958664969011005421358017