L(s) = 1 | + (0.629 + 0.777i)2-s + (−0.777 − 0.629i)3-s + (−0.207 + 0.978i)4-s − i·6-s + (0.284 + 0.958i)7-s + (−0.891 + 0.453i)8-s + (0.207 + 0.978i)9-s + (−0.958 + 0.284i)11-s + (0.777 − 0.629i)12-s + (0.999 − 0.0261i)13-s + (−0.566 + 0.824i)14-s + (−0.913 − 0.406i)16-s + (0.972 − 0.233i)17-s + (−0.629 + 0.777i)18-s + (−0.824 + 0.566i)19-s + ⋯ |
L(s) = 1 | + (0.629 + 0.777i)2-s + (−0.777 − 0.629i)3-s + (−0.207 + 0.978i)4-s − i·6-s + (0.284 + 0.958i)7-s + (−0.891 + 0.453i)8-s + (0.207 + 0.978i)9-s + (−0.958 + 0.284i)11-s + (0.777 − 0.629i)12-s + (0.999 − 0.0261i)13-s + (−0.566 + 0.824i)14-s + (−0.913 − 0.406i)16-s + (0.972 − 0.233i)17-s + (−0.629 + 0.777i)18-s + (−0.824 + 0.566i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3754157065 - 0.2407609573i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3754157065 - 0.2407609573i\) |
\(L(1)\) |
\(\approx\) |
\(0.8614789543 + 0.3794984737i\) |
\(L(1)\) |
\(\approx\) |
\(0.8614789543 + 0.3794984737i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 \) |
good | 2 | \( 1 + (0.629 + 0.777i)T \) |
| 3 | \( 1 + (-0.777 - 0.629i)T \) |
| 7 | \( 1 + (0.284 + 0.958i)T \) |
| 11 | \( 1 + (-0.958 + 0.284i)T \) |
| 13 | \( 1 + (0.999 - 0.0261i)T \) |
| 17 | \( 1 + (0.972 - 0.233i)T \) |
| 19 | \( 1 + (-0.824 + 0.566i)T \) |
| 23 | \( 1 + (-0.0784 - 0.996i)T \) |
| 29 | \( 1 + (-0.629 - 0.777i)T \) |
| 31 | \( 1 + (0.725 - 0.688i)T \) |
| 37 | \( 1 + (-0.333 + 0.942i)T \) |
| 41 | \( 1 + (0.707 + 0.707i)T \) |
| 43 | \( 1 + (-0.233 - 0.972i)T \) |
| 47 | \( 1 + (0.156 + 0.987i)T \) |
| 53 | \( 1 + (0.358 - 0.933i)T \) |
| 59 | \( 1 + (-0.998 + 0.0523i)T \) |
| 61 | \( 1 + (-0.453 - 0.891i)T \) |
| 67 | \( 1 + (-0.933 + 0.358i)T \) |
| 71 | \( 1 + (-0.878 - 0.477i)T \) |
| 73 | \( 1 + (-0.522 + 0.852i)T \) |
| 79 | \( 1 + (0.707 + 0.707i)T \) |
| 83 | \( 1 + (0.104 - 0.994i)T \) |
| 89 | \( 1 + (-0.878 - 0.477i)T \) |
| 97 | \( 1 + (-0.913 + 0.406i)T \) |
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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
−17.92535618720215235768751644387, −17.21423572507799567627530114534, −16.371793513573607803447183609882, −15.89261729318890868438418037703, −15.14575384669662878995209469029, −14.558990648165347519730005783789, −13.65443533312015149567288291660, −13.30381631552040836767059573438, −12.46829714630687382036665615754, −11.86328332676780863217685767942, −10.94358898937360816937245529205, −10.7472404951638652854635281137, −10.3044831665118750749509491903, −9.4065984138786380680071082906, −8.72969197559217855289564446770, −7.69244462740076571820906499940, −6.86370621028265219756460680268, −5.99411980912485562104002446196, −5.5024925604227263924488355338, −4.81694692687872375155165567843, −4.07833121636184945051421820169, −3.557960245741379599495102248104, −2.845550155683233899229080065720, −1.55344307895310641373624533957, −0.99192379650455536609687706098,
0.10899988034190450647142764690, 1.49678175136544269770352579453, 2.39039215378729786614759465816, 3.025079315262762929303592035227, 4.214455181896313812170088092120, 4.79620274707329556115942294701, 5.63734339485935241235900672454, 5.95263814878795608266976576735, 6.56201804526820493430960327794, 7.49728729255862474092071426549, 8.164308045132290339702742945981, 8.39897514839538417404590181998, 9.55652640218752218831986719606, 10.46442525376929939509769033590, 11.21284463665460588948982658416, 11.91915048210283651111141905567, 12.40149081639019834817174097655, 12.998591289790621599643973461075, 13.54544780767614740208997838133, 14.32642310181600148235423632863, 15.09200238658315275681855794763, 15.61048573534438374661143467789, 16.30454154854834801556209986386, 16.81797191427317487941430567236, 17.54892334582309986938634718931