L(s) = 1 | + (0.809 + 0.587i)3-s + (0.309 − 0.951i)5-s + (0.809 − 0.587i)7-s + (0.309 + 0.951i)9-s + (0.309 + 0.951i)13-s + (0.809 − 0.587i)15-s + (0.309 − 0.951i)17-s + (0.809 + 0.587i)19-s + 21-s − 23-s + (−0.809 − 0.587i)25-s + (−0.309 + 0.951i)27-s + (−0.809 + 0.587i)29-s + (−0.309 − 0.951i)31-s + (−0.309 − 0.951i)35-s + ⋯ |
L(s) = 1 | + (0.809 + 0.587i)3-s + (0.309 − 0.951i)5-s + (0.809 − 0.587i)7-s + (0.309 + 0.951i)9-s + (0.309 + 0.951i)13-s + (0.809 − 0.587i)15-s + (0.309 − 0.951i)17-s + (0.809 + 0.587i)19-s + 21-s − 23-s + (−0.809 − 0.587i)25-s + (−0.309 + 0.951i)27-s + (−0.809 + 0.587i)29-s + (−0.309 − 0.951i)31-s + (−0.309 − 0.951i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.120847620 + 0.02521161975i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.120847620 + 0.02521161975i\) |
\(L(1)\) |
\(\approx\) |
\(1.532210475 + 0.03643351569i\) |
\(L(1)\) |
\(\approx\) |
\(1.532210475 + 0.03643351569i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + (0.809 + 0.587i)T \) |
| 5 | \( 1 + (0.309 - 0.951i)T \) |
| 7 | \( 1 + (0.809 - 0.587i)T \) |
| 13 | \( 1 + (0.309 + 0.951i)T \) |
| 17 | \( 1 + (0.309 - 0.951i)T \) |
| 19 | \( 1 + (0.809 + 0.587i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (-0.809 + 0.587i)T \) |
| 31 | \( 1 + (-0.309 - 0.951i)T \) |
| 37 | \( 1 + (-0.809 + 0.587i)T \) |
| 41 | \( 1 + (-0.809 - 0.587i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (0.809 + 0.587i)T \) |
| 53 | \( 1 + (0.309 + 0.951i)T \) |
| 59 | \( 1 + (0.809 - 0.587i)T \) |
| 61 | \( 1 + (0.309 - 0.951i)T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + (-0.309 + 0.951i)T \) |
| 73 | \( 1 + (-0.809 + 0.587i)T \) |
| 79 | \( 1 + (-0.309 - 0.951i)T \) |
| 83 | \( 1 + (-0.309 + 0.951i)T \) |
| 89 | \( 1 + T \) |
| 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
−34.39878985917838826041223035158, −32.99043700794769474473874003342, −31.69028417376710656480657364379, −30.40810113878920278753504308418, −30.11690357671091393808131912951, −28.39240728581687361433391055341, −26.918052485463230726723885058443, −25.84327288265596404624485793559, −24.89744075359983059199875876713, −23.7299652207707264093112472389, −22.18061683998117381433338267447, −20.99187035358729752890035162497, −19.67005016983163275871124655102, −18.3729178796391340823330463949, −17.73702413900077357848978999070, −15.36964054976571955614179883777, −14.533777685953885593950040618340, −13.35250636932079065766103042274, −11.81437440968172593052911125476, −10.23305919263583406651242412549, −8.573244594583036803292576545267, −7.39337960307907603264948281719, −5.80291389716235351742311101179, −3.35313140344065141300371893875, −1.89128691692274294189259062066,
1.70517376550139498620544639056, 3.94540582846430890848593247677, 5.186039188173345398219983388094, 7.61765554963716825429929031663, 8.8657363571163944846817778988, 10.01247109345651253294879116836, 11.7005337041833146223390914089, 13.54530925416870274614199620723, 14.30681670669714702161839578699, 15.97947976546516004486151035897, 16.913362738862897498169780715053, 18.584753854340232542888776550345, 20.34039671226209999924523330444, 20.6693684829597290560639306736, 21.97694340279532990810005258908, 23.821340555279006647927917085414, 24.77819160688995790915976734729, 26.01505327701698906833488030972, 27.17145833024310271749714268253, 28.12874271153399275142878040697, 29.59270681672405290588600650773, 31.01837581008879917284198753925, 31.84083252375502723146456049261, 33.09471834866611544305101885476, 33.72335995347242087506035851772