L(s) = 1 | + (−0.119 + 0.992i)2-s + (0.894 + 0.447i)3-s + (−0.971 − 0.237i)4-s + (−0.550 + 0.834i)6-s + (0.351 − 0.936i)8-s + (0.599 + 0.800i)9-s + (−0.599 + 0.800i)11-s + (−0.762 − 0.646i)12-s + (−0.657 − 0.753i)13-s + (0.887 + 0.460i)16-s + (−0.959 − 0.280i)17-s + (−0.866 + 0.5i)18-s + (0.978 + 0.207i)19-s + (−0.722 − 0.691i)22-s + (0.941 + 0.337i)23-s + (0.733 − 0.680i)24-s + ⋯ |
L(s) = 1 | + (−0.119 + 0.992i)2-s + (0.894 + 0.447i)3-s + (−0.971 − 0.237i)4-s + (−0.550 + 0.834i)6-s + (0.351 − 0.936i)8-s + (0.599 + 0.800i)9-s + (−0.599 + 0.800i)11-s + (−0.762 − 0.646i)12-s + (−0.657 − 0.753i)13-s + (0.887 + 0.460i)16-s + (−0.959 − 0.280i)17-s + (−0.866 + 0.5i)18-s + (0.978 + 0.207i)19-s + (−0.722 − 0.691i)22-s + (0.941 + 0.337i)23-s + (0.733 − 0.680i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.895552062 + 0.6916071713i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.895552062 + 0.6916071713i\) |
\(L(1)\) |
\(\approx\) |
\(1.001344927 + 0.5885364678i\) |
\(L(1)\) |
\(\approx\) |
\(1.001344927 + 0.5885364678i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.119 + 0.992i)T \) |
| 3 | \( 1 + (0.894 + 0.447i)T \) |
| 11 | \( 1 + (-0.599 + 0.800i)T \) |
| 13 | \( 1 + (-0.657 - 0.753i)T \) |
| 17 | \( 1 + (-0.959 - 0.280i)T \) |
| 19 | \( 1 + (0.978 + 0.207i)T \) |
| 23 | \( 1 + (0.941 + 0.337i)T \) |
| 29 | \( 1 + (-0.134 - 0.990i)T \) |
| 31 | \( 1 + (0.669 - 0.743i)T \) |
| 37 | \( 1 + (-0.762 - 0.646i)T \) |
| 41 | \( 1 + (-0.963 + 0.266i)T \) |
| 43 | \( 1 + (0.781 - 0.623i)T \) |
| 47 | \( 1 + (-0.486 - 0.873i)T \) |
| 53 | \( 1 + (0.237 - 0.971i)T \) |
| 59 | \( 1 + (0.842 - 0.538i)T \) |
| 61 | \( 1 + (0.525 + 0.850i)T \) |
| 67 | \( 1 + (0.743 + 0.669i)T \) |
| 71 | \( 1 + (-0.691 + 0.722i)T \) |
| 73 | \( 1 + (-0.981 - 0.193i)T \) |
| 79 | \( 1 + (-0.669 - 0.743i)T \) |
| 83 | \( 1 + (-0.512 - 0.858i)T \) |
| 89 | \( 1 + (0.946 - 0.323i)T \) |
| 97 | \( 1 + (0.951 + 0.309i)T \) |
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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
−20.71366828364474929076252979414, −20.079734374060291986203324150511, −19.28130355403846947371382651722, −18.87161346154418500695359041778, −18.06341514182772940776839188703, −17.34708303929370830237652723758, −16.26739813532987140581644722007, −15.29959141830511552579528152912, −14.29720506537700117135185460432, −13.79071140312294548183303830135, −13.066294120177613267844881089019, −12.361554709387946079825138561351, −11.49916229326965329871063520295, −10.678211751723219833596541915685, −9.76744581981581351120805222164, −8.937709538745443322184563936992, −8.48525941449972134282477276848, −7.47054467807501450126041850731, −6.63136180778156366800289754081, −5.20650529795033534325807243373, −4.39522552347344108981320251602, −3.251231332145640675671497303206, −2.764425951534515755523382928428, −1.756053901266065855108419851157, −0.83838519658580476540587174037,
0.457016903427381924087814299206, 2.00676621038722091347882230248, 3.02612412596969925757496819027, 4.08596225604625355743058922689, 4.92592022256179683580380304288, 5.52897789909262167660684065808, 6.993818254973126926529602224547, 7.4490195247226856769172967157, 8.26731896272149906031029784892, 9.05265897015804037804278030225, 9.91769899105888437974256748623, 10.27811718074613507711225510522, 11.682450079337663008575274170698, 13.05645892022669062144466327884, 13.27231264157371979613722263358, 14.30947125764763655133371507144, 15.0309312054687082512146881042, 15.54050761892832908963483863621, 16.11115366982216394241058481904, 17.256782269331701366471782062539, 17.76082427487500386744625840181, 18.729702108754225823225242899232, 19.40831864794595159674964615131, 20.30396649105975547384316743786, 20.8951213793165407599630851772