L(s) = 1 | + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (0.5 + 0.866i)5-s + (−0.978 − 0.207i)7-s + (−0.309 + 0.951i)8-s + (−0.104 + 0.994i)10-s + (0.669 − 0.743i)11-s + (−0.913 − 0.406i)13-s + (−0.669 − 0.743i)14-s + (−0.809 + 0.587i)16-s + (0.669 + 0.743i)17-s + (0.913 − 0.406i)19-s + (−0.669 + 0.743i)20-s + (0.978 − 0.207i)22-s + (0.309 − 0.951i)23-s + ⋯ |
L(s) = 1 | + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (0.5 + 0.866i)5-s + (−0.978 − 0.207i)7-s + (−0.309 + 0.951i)8-s + (−0.104 + 0.994i)10-s + (0.669 − 0.743i)11-s + (−0.913 − 0.406i)13-s + (−0.669 − 0.743i)14-s + (−0.809 + 0.587i)16-s + (0.669 + 0.743i)17-s + (0.913 − 0.406i)19-s + (−0.669 + 0.743i)20-s + (0.978 − 0.207i)22-s + (0.309 − 0.951i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.147253220 + 0.9481605123i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.147253220 + 0.9481605123i\) |
\(L(1)\) |
\(\approx\) |
\(1.325382442 + 0.6915264188i\) |
\(L(1)\) |
\(\approx\) |
\(1.325382442 + 0.6915264188i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.978 - 0.207i)T \) |
| 11 | \( 1 + (0.669 - 0.743i)T \) |
| 13 | \( 1 + (-0.913 - 0.406i)T \) |
| 17 | \( 1 + (0.669 + 0.743i)T \) |
| 19 | \( 1 + (0.913 - 0.406i)T \) |
| 23 | \( 1 + (0.309 - 0.951i)T \) |
| 29 | \( 1 + (-0.809 - 0.587i)T \) |
| 37 | \( 1 + (0.5 - 0.866i)T \) |
| 41 | \( 1 + (0.104 - 0.994i)T \) |
| 43 | \( 1 + (-0.913 + 0.406i)T \) |
| 47 | \( 1 + (0.809 - 0.587i)T \) |
| 53 | \( 1 + (-0.978 + 0.207i)T \) |
| 59 | \( 1 + (0.104 + 0.994i)T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.978 - 0.207i)T \) |
| 73 | \( 1 + (-0.669 + 0.743i)T \) |
| 79 | \( 1 + (-0.669 - 0.743i)T \) |
| 83 | \( 1 + (-0.104 + 0.994i)T \) |
| 89 | \( 1 + (0.309 + 0.951i)T \) |
| 97 | \( 1 + (0.309 + 0.951i)T \) |
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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
−29.88999682126489658539432669668, −29.13168663797249887849706648362, −28.37684411346419750187722545977, −27.280873470272622657933285645888, −25.44772524671588738766693510460, −24.8270700637013856940062114719, −23.623687093755778832426065105513, −22.48905020353146807487211536239, −21.73509052412672007144030868906, −20.47599647848809198938878988070, −19.79212148636880679348961634169, −18.591880033507568869820674997725, −16.984063044614473014573782083403, −15.949724624483628857105102883658, −14.607842045909079826454964636259, −13.500640058523342391297475834552, −12.482342993232149681333467552256, −11.75365482868840024264460902254, −9.72369210737143371461784207589, −9.50239788117495179024901506291, −7.11804957567405439848525030200, −5.74354075273294646928295029238, −4.67108647368835285595237293573, −3.1464399771849523203599602592, −1.55200013428044127451767819918,
2.713931073056768087936751565746, 3.72832545061314768916565993719, 5.591532445609509996461832458490, 6.52513483224548923836954688037, 7.570406327646563305364846633288, 9.32517635263457056730302875296, 10.68871150168001399456484214869, 12.11122085599234731378076206894, 13.27595965114938206578066889868, 14.24901358721086692237456907063, 15.158025550298454362361833398672, 16.494828217841550745294626466394, 17.310196902344377239932035262458, 18.77417780235760625487736592532, 19.97084160258863267450030160468, 21.48557037166111198754458505124, 22.28024797604471756397563101046, 22.90222418066703702254459955013, 24.32154876993843096440666250971, 25.21189625077995128146534969808, 26.24289953384354962089171912392, 26.89520796488944874696638166980, 28.82970264448318361835692648804, 29.8426795450523044334788299963, 30.37172222543553783127759379329