L(s) = 1 | + (0.739 − 0.673i)2-s + (−0.850 + 0.526i)3-s + (0.0922 − 0.995i)4-s + (−0.602 + 0.798i)5-s + (−0.273 + 0.961i)6-s + (0.932 + 0.361i)7-s + (−0.602 − 0.798i)8-s + (0.445 − 0.895i)9-s + (0.0922 + 0.995i)10-s + (0.932 + 0.361i)11-s + (0.445 + 0.895i)12-s + (0.445 − 0.895i)13-s + (0.932 − 0.361i)14-s + (0.0922 − 0.995i)15-s + (−0.982 − 0.183i)16-s + (−0.273 + 0.961i)17-s + ⋯ |
L(s) = 1 | + (0.739 − 0.673i)2-s + (−0.850 + 0.526i)3-s + (0.0922 − 0.995i)4-s + (−0.602 + 0.798i)5-s + (−0.273 + 0.961i)6-s + (0.932 + 0.361i)7-s + (−0.602 − 0.798i)8-s + (0.445 − 0.895i)9-s + (0.0922 + 0.995i)10-s + (0.932 + 0.361i)11-s + (0.445 + 0.895i)12-s + (0.445 − 0.895i)13-s + (0.932 − 0.361i)14-s + (0.0922 − 0.995i)15-s + (−0.982 − 0.183i)16-s + (−0.273 + 0.961i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 953 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 953 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.702762399 - 0.3545780470i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.702762399 - 0.3545780470i\) |
\(L(1)\) |
\(\approx\) |
\(1.257608427 - 0.2174346367i\) |
\(L(1)\) |
\(\approx\) |
\(1.257608427 - 0.2174346367i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 953 | \( 1 \) |
good | 2 | \( 1 + (0.739 - 0.673i)T \) |
| 3 | \( 1 + (-0.850 + 0.526i)T \) |
| 5 | \( 1 + (-0.602 + 0.798i)T \) |
| 7 | \( 1 + (0.932 + 0.361i)T \) |
| 11 | \( 1 + (0.932 + 0.361i)T \) |
| 13 | \( 1 + (0.445 - 0.895i)T \) |
| 17 | \( 1 + (-0.273 + 0.961i)T \) |
| 19 | \( 1 + (0.445 - 0.895i)T \) |
| 23 | \( 1 + (-0.982 + 0.183i)T \) |
| 29 | \( 1 + (-0.273 + 0.961i)T \) |
| 31 | \( 1 + (0.445 - 0.895i)T \) |
| 37 | \( 1 + (0.739 + 0.673i)T \) |
| 41 | \( 1 + (0.0922 - 0.995i)T \) |
| 43 | \( 1 + (0.739 + 0.673i)T \) |
| 47 | \( 1 + (-0.273 - 0.961i)T \) |
| 53 | \( 1 + (-0.273 + 0.961i)T \) |
| 59 | \( 1 + (-0.273 + 0.961i)T \) |
| 61 | \( 1 + (0.932 - 0.361i)T \) |
| 67 | \( 1 + (0.445 - 0.895i)T \) |
| 71 | \( 1 + (0.445 - 0.895i)T \) |
| 73 | \( 1 + (0.932 - 0.361i)T \) |
| 79 | \( 1 + (0.445 + 0.895i)T \) |
| 83 | \( 1 + (0.932 - 0.361i)T \) |
| 89 | \( 1 + (-0.602 - 0.798i)T \) |
| 97 | \( 1 + (0.932 + 0.361i)T \) |
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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
−22.06471294277888758026005095758, −21.1431503039653841225932547281, −20.517160856762708606133622703356, −19.518359052416211704825893547569, −18.46041463330797839488555895714, −17.63421595699312066358492478091, −16.955726199866960535511838836256, −16.231837782193107760614132888910, −15.87089302261847404218707333724, −14.39461287492820066039648969638, −13.98005883125047433704931592892, −13.07618894021008389468756016840, −12.08661203093720387391274864549, −11.64654088818793627135695069170, −11.14137100568377544882063604860, −9.48130655392777899151730314530, −8.34942608131515071761544692495, −7.80813877147368683982350879614, −6.91435747224966261374319079806, −6.08328824551718536299008002016, −5.18572535919571306436966158917, −4.38795779509453827005937438407, −3.829921983330214275122645384306, −2.02884207906503494704063720707, −0.94570641875665324846680883223,
0.93409186873973829825586731933, 2.140515605412121852835379718309, 3.410692849645455510631241599453, 4.09605358659605283853878127708, 4.87916556389200774376742092038, 5.885907430992218281040332939637, 6.50957857161815786923558241737, 7.652218357486260155919318382980, 8.92872364872969311860693156617, 9.9620553000607486350920871332, 10.795909302633617681854324510762, 11.26306648697572265299948899099, 11.92690441577321735350710754206, 12.612817431266047364920056903339, 13.83037525064128472712624886252, 14.77632662774805078007572061181, 15.22043728191666556888981384995, 15.80313299646224252600787298147, 17.137061149467455296691130703114, 18.01118093506736378370034567842, 18.4402943453071520930916942713, 19.67492327508170440301321255892, 20.20394272835782702913576661707, 21.156005014195980087429136723598, 22.047052408631817861434393415346