L(s) = 1 | + (0.104 − 0.994i)2-s + (−0.978 − 0.207i)3-s + (−0.978 − 0.207i)4-s + (−0.309 + 0.951i)6-s + (−0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (−0.913 + 0.406i)11-s + (0.913 + 0.406i)12-s + (−0.809 + 0.587i)13-s + (0.913 + 0.406i)16-s + (−0.669 − 0.743i)17-s + (0.5 − 0.866i)18-s + (−0.978 + 0.207i)19-s + (0.309 + 0.951i)22-s + (0.5 − 0.866i)24-s + ⋯ |
L(s) = 1 | + (0.104 − 0.994i)2-s + (−0.978 − 0.207i)3-s + (−0.978 − 0.207i)4-s + (−0.309 + 0.951i)6-s + (−0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (−0.913 + 0.406i)11-s + (0.913 + 0.406i)12-s + (−0.809 + 0.587i)13-s + (0.913 + 0.406i)16-s + (−0.669 − 0.743i)17-s + (0.5 − 0.866i)18-s + (−0.978 + 0.207i)19-s + (0.309 + 0.951i)22-s + (0.5 − 0.866i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.02943113214 - 0.1985606323i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02943113214 - 0.1985606323i\) |
\(L(1)\) |
\(\approx\) |
\(0.5072688594 - 0.2613171870i\) |
\(L(1)\) |
\(\approx\) |
\(0.5072688594 - 0.2613171870i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + (0.104 - 0.994i)T \) |
| 3 | \( 1 + (-0.978 - 0.207i)T \) |
| 11 | \( 1 + (-0.913 + 0.406i)T \) |
| 13 | \( 1 + (-0.809 + 0.587i)T \) |
| 17 | \( 1 + (-0.669 - 0.743i)T \) |
| 19 | \( 1 + (-0.978 + 0.207i)T \) |
| 29 | \( 1 + (0.309 + 0.951i)T \) |
| 31 | \( 1 + (-0.669 - 0.743i)T \) |
| 37 | \( 1 + (0.913 + 0.406i)T \) |
| 41 | \( 1 + (0.809 - 0.587i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (0.669 - 0.743i)T \) |
| 53 | \( 1 + (-0.978 - 0.207i)T \) |
| 59 | \( 1 + (0.104 + 0.994i)T \) |
| 61 | \( 1 + (-0.104 + 0.994i)T \) |
| 67 | \( 1 + (0.669 + 0.743i)T \) |
| 71 | \( 1 + (0.309 + 0.951i)T \) |
| 73 | \( 1 + (0.913 - 0.406i)T \) |
| 79 | \( 1 + (-0.669 + 0.743i)T \) |
| 83 | \( 1 + (-0.309 + 0.951i)T \) |
| 89 | \( 1 + (-0.104 + 0.994i)T \) |
| 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
−18.65482649843919806196329475959, −17.78484756053749519115628140424, −17.42249005493001802718174047865, −16.89515201748778936582027095164, −15.9775747446501246969816102474, −15.66673107319488782354753914713, −14.957914489136108154199034105873, −14.276202072089285094388354636099, −13.22636771137454238913817291586, −12.75648224344944608198815845335, −12.31021720758247810180999788432, −11.01370200365979541975626527323, −10.7043235023809418404633155401, −9.784450481402199276038393067597, −9.16419089881074369084936933845, −8.11340379777318721452238328084, −7.67911266093532671507240883773, −6.74890925871031248190263033079, −6.095966782700213380412054973560, −5.588953372192580283488702843707, −4.70891818381716149354945242054, −4.32183254764165841207410320531, −3.26163156909648735125722650440, −2.166088980341807218736008381296, −0.72458725429172567463566161488,
0.09703036520151514834366319388, 1.15366298267847031393420805943, 2.26533995772104674865679154230, 2.52994973025986023527467079959, 4.0119997269989431014546810460, 4.49660273498751182005303269210, 5.21230546623840130711043184974, 5.84472679993081208750145905306, 6.88907492316704993356208432432, 7.50402891893753183749913828774, 8.46685508724963486799795312007, 9.4014445407126316462490092624, 9.936406615256117773184600224188, 10.78090602872978818274238771827, 11.11244281761670119848287509432, 11.95496715695777425224206328108, 12.56339360992349207185490717613, 12.993961520339782189395490524109, 13.76455966734707545257764925466, 14.59306720213745512452405751085, 15.35760435165394024178201246715, 16.220656720227928851658308479076, 16.95549422709186034372465372428, 17.564984283731951725157500926718, 18.24294710292264457165773633723