L(s) = 1 | + (−0.415 − 0.909i)2-s + (−0.654 + 0.755i)4-s + (−0.841 − 0.540i)5-s + (−0.142 − 0.989i)7-s + (0.959 + 0.281i)8-s + (−0.142 + 0.989i)10-s + (−0.415 + 0.909i)11-s + (−0.142 + 0.989i)13-s + (−0.841 + 0.540i)14-s + (−0.142 − 0.989i)16-s + (0.654 + 0.755i)17-s + (−0.654 + 0.755i)19-s + (0.959 − 0.281i)20-s + 22-s + (0.415 + 0.909i)25-s + (0.959 − 0.281i)26-s + ⋯ |
L(s) = 1 | + (−0.415 − 0.909i)2-s + (−0.654 + 0.755i)4-s + (−0.841 − 0.540i)5-s + (−0.142 − 0.989i)7-s + (0.959 + 0.281i)8-s + (−0.142 + 0.989i)10-s + (−0.415 + 0.909i)11-s + (−0.142 + 0.989i)13-s + (−0.841 + 0.540i)14-s + (−0.142 − 0.989i)16-s + (0.654 + 0.755i)17-s + (−0.654 + 0.755i)19-s + (0.959 − 0.281i)20-s + 22-s + (0.415 + 0.909i)25-s + (0.959 − 0.281i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3045172176 + 0.2045272664i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3045172176 + 0.2045272664i\) |
\(L(1)\) |
\(\approx\) |
\(0.5464928499 - 0.1738640890i\) |
\(L(1)\) |
\(\approx\) |
\(0.5464928499 - 0.1738640890i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + (-0.415 - 0.909i)T \) |
| 5 | \( 1 + (-0.841 - 0.540i)T \) |
| 7 | \( 1 + (-0.142 - 0.989i)T \) |
| 11 | \( 1 + (-0.415 + 0.909i)T \) |
| 13 | \( 1 + (-0.142 + 0.989i)T \) |
| 17 | \( 1 + (0.654 + 0.755i)T \) |
| 19 | \( 1 + (-0.654 + 0.755i)T \) |
| 29 | \( 1 + (0.654 + 0.755i)T \) |
| 31 | \( 1 + (-0.959 - 0.281i)T \) |
| 37 | \( 1 + (0.841 - 0.540i)T \) |
| 41 | \( 1 + (-0.841 - 0.540i)T \) |
| 43 | \( 1 + (-0.959 + 0.281i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (0.142 + 0.989i)T \) |
| 59 | \( 1 + (0.142 - 0.989i)T \) |
| 61 | \( 1 + (-0.959 - 0.281i)T \) |
| 67 | \( 1 + (0.415 + 0.909i)T \) |
| 71 | \( 1 + (-0.415 - 0.909i)T \) |
| 73 | \( 1 + (-0.654 + 0.755i)T \) |
| 79 | \( 1 + (-0.142 + 0.989i)T \) |
| 83 | \( 1 + (-0.841 + 0.540i)T \) |
| 89 | \( 1 + (0.959 - 0.281i)T \) |
| 97 | \( 1 + (0.841 + 0.540i)T \) |
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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
−31.802569788669165042740451938565, −30.47782183154511796218874282370, −28.9665340610073924987104716209, −27.693461091087701639902313641378, −27.07483313184785701723578051468, −25.85880446470623031440137295115, −24.90718513739537517694471011675, −23.7666653118581111686455438420, −22.79369239719686826452000080137, −21.71601441798008427200318550198, −19.785640867634808473162366602478, −18.785000472068479918710651767795, −18.050118707811245758762878446624, −16.43100138893861229251062512207, −15.49847847540921829593689479496, −14.723491274945997063596600032046, −13.18680701721911723398250867270, −11.564063519780019386559445653468, −10.21537926475686330760926569504, −8.67604187554937166889866630620, −7.75498121407437010944945089673, −6.3298243903386481267019363925, −5.056692032962853973567038238963, −3.046254424380743597183650843539, −0.22272967373836176981798339233,
1.56636609866453987841111443876, 3.67657999812304108789888703834, 4.62542020349987665815092429613, 7.23094474738210972482947068487, 8.30246201793445477897299218667, 9.7458547899604849891909930275, 10.85719252664905441240859192941, 12.165093631623655888542730315157, 13.00739863294234763932311057408, 14.5428302702462413862018051204, 16.3618273451180834328118621106, 17.110933267882072931612320822557, 18.61590368935399954731173534340, 19.678817049620107408537956995732, 20.42864103007609278700660870349, 21.49983655275671384270895861906, 23.11192626242612301865944032573, 23.649935592041386878012920662124, 25.58548888501943513456978573744, 26.60719191309940784894538391921, 27.56458291245267139937318214134, 28.49198212921696127324084956286, 29.52578841590835137619285651632, 30.702985412136030318968070900325, 31.45182203697803533078267991602