L(s) = 1 | + (0.425 + 0.904i)2-s + (−0.0627 + 0.998i)3-s + (−0.637 + 0.770i)4-s + (−0.929 + 0.368i)6-s + (−0.309 + 0.951i)7-s + (−0.968 − 0.248i)8-s + (−0.992 − 0.125i)9-s + (−0.425 − 0.904i)11-s + (−0.728 − 0.684i)12-s + (0.992 + 0.125i)13-s + (−0.992 + 0.125i)14-s + (−0.187 − 0.982i)16-s + (0.637 + 0.770i)17-s + (−0.309 − 0.951i)18-s + (0.0627 + 0.998i)19-s + ⋯ |
L(s) = 1 | + (0.425 + 0.904i)2-s + (−0.0627 + 0.998i)3-s + (−0.637 + 0.770i)4-s + (−0.929 + 0.368i)6-s + (−0.309 + 0.951i)7-s + (−0.968 − 0.248i)8-s + (−0.992 − 0.125i)9-s + (−0.425 − 0.904i)11-s + (−0.728 − 0.684i)12-s + (0.992 + 0.125i)13-s + (−0.992 + 0.125i)14-s + (−0.187 − 0.982i)16-s + (0.637 + 0.770i)17-s + (−0.309 − 0.951i)18-s + (0.0627 + 0.998i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0502i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0502i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.02593268249 + 1.031611389i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02593268249 + 1.031611389i\) |
\(L(1)\) |
\(\approx\) |
\(0.5958900380 + 0.8822730763i\) |
\(L(1)\) |
\(\approx\) |
\(0.5958900380 + 0.8822730763i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 + (0.425 + 0.904i)T \) |
| 3 | \( 1 + (-0.0627 + 0.998i)T \) |
| 7 | \( 1 + (-0.309 + 0.951i)T \) |
| 11 | \( 1 + (-0.425 - 0.904i)T \) |
| 13 | \( 1 + (0.992 + 0.125i)T \) |
| 17 | \( 1 + (0.637 + 0.770i)T \) |
| 19 | \( 1 + (0.0627 + 0.998i)T \) |
| 23 | \( 1 + (-0.876 - 0.481i)T \) |
| 29 | \( 1 + (0.535 + 0.844i)T \) |
| 31 | \( 1 + (-0.637 - 0.770i)T \) |
| 37 | \( 1 + (0.187 + 0.982i)T \) |
| 41 | \( 1 + (0.876 - 0.481i)T \) |
| 43 | \( 1 + (0.809 + 0.587i)T \) |
| 47 | \( 1 + (-0.968 + 0.248i)T \) |
| 53 | \( 1 + (0.929 + 0.368i)T \) |
| 59 | \( 1 + (0.728 + 0.684i)T \) |
| 61 | \( 1 + (0.876 + 0.481i)T \) |
| 67 | \( 1 + (-0.535 + 0.844i)T \) |
| 71 | \( 1 + (0.968 - 0.248i)T \) |
| 73 | \( 1 + (-0.728 + 0.684i)T \) |
| 79 | \( 1 + (0.0627 - 0.998i)T \) |
| 83 | \( 1 + (-0.0627 - 0.998i)T \) |
| 89 | \( 1 + (0.728 - 0.684i)T \) |
| 97 | \( 1 + (-0.535 - 0.844i)T \) |
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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
−28.62603734252733865585977739154, −27.85049507186575448408593398468, −26.37217456884153851788076993664, −25.36671476348725973545425110957, −23.96383437438642246874382861316, −23.22828336554172819054084521839, −22.67392594009520946353332655671, −21.09303631280732455168788878737, −20.12989335703127647735191022424, −19.47395402016753109128992177883, −18.20610627776392342745984744609, −17.61467950334755827067072883034, −15.942340849016031529150454776581, −14.34473866034581163347427716666, −13.508118677225205310286905397550, −12.76783502789384585689483780967, −11.6477999245048671061420289274, −10.61513564506624244953001947729, −9.3970213386925970262287279015, −7.82904887226903989616713270422, −6.61349714628697256872555835329, −5.24853559917938827621666132979, −3.73619480686890788191466485464, −2.35280003364568684221290676905, −0.887857656936832619916044497070,
3.057226800601453913289280737705, 4.07157686785020646693928173755, 5.651626007646838249501265562131, 6.05442677279285443609242519087, 8.15881633544301335417370874026, 8.83136545354960794541398460676, 10.16006487503325358167038398759, 11.597648987239847888522815253334, 12.80350976930608106630914747013, 14.13917072196322506166091447124, 14.996680581687088768739256441395, 16.154829266560665678110541031932, 16.406747678023889531974158871204, 17.96414516723501406332914110507, 18.95835983683041203275263537974, 20.769688070625839539442476488557, 21.51590817117196845078860955617, 22.30833579094618414260886235260, 23.292569106571360228948685492344, 24.35863287260889643773497696964, 25.672575281342217329960572671734, 26.05149530767384073357851919553, 27.33169494608014315096288105005, 28.05276996130186240498676975426, 29.29795227493933522437022831078