L(s) = 1 | + (−0.342 − 0.939i)3-s + (−0.984 − 0.173i)7-s + (−0.766 + 0.642i)9-s + (−0.5 − 0.866i)11-s + (−0.766 − 0.642i)13-s + (−0.766 + 0.642i)17-s + (0.342 + 0.939i)19-s + (0.173 + 0.984i)21-s + (−0.5 + 0.866i)23-s + (0.866 + 0.5i)27-s + (0.866 − 0.5i)29-s + i·31-s + (−0.642 + 0.766i)33-s + (−0.342 + 0.939i)39-s + (−0.766 − 0.642i)41-s + ⋯ |
L(s) = 1 | + (−0.342 − 0.939i)3-s + (−0.984 − 0.173i)7-s + (−0.766 + 0.642i)9-s + (−0.5 − 0.866i)11-s + (−0.766 − 0.642i)13-s + (−0.766 + 0.642i)17-s + (0.342 + 0.939i)19-s + (0.173 + 0.984i)21-s + (−0.5 + 0.866i)23-s + (0.866 + 0.5i)27-s + (0.866 − 0.5i)29-s + i·31-s + (−0.642 + 0.766i)33-s + (−0.342 + 0.939i)39-s + (−0.766 − 0.642i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6766291143 + 0.01817324453i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6766291143 + 0.01817324453i\) |
\(L(1)\) |
\(\approx\) |
\(0.6751026271 - 0.1853353447i\) |
\(L(1)\) |
\(\approx\) |
\(0.6751026271 - 0.1853353447i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 3 | \( 1 + (-0.342 - 0.939i)T \) |
| 7 | \( 1 + (-0.984 - 0.173i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.766 - 0.642i)T \) |
| 17 | \( 1 + (-0.766 + 0.642i)T \) |
| 19 | \( 1 + (0.342 + 0.939i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.866 - 0.5i)T \) |
| 31 | \( 1 + iT \) |
| 41 | \( 1 + (-0.766 - 0.642i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (0.866 + 0.5i)T \) |
| 53 | \( 1 + (0.984 - 0.173i)T \) |
| 59 | \( 1 + (0.984 - 0.173i)T \) |
| 61 | \( 1 + (0.642 - 0.766i)T \) |
| 67 | \( 1 + (-0.984 - 0.173i)T \) |
| 71 | \( 1 + (-0.939 + 0.342i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (0.984 + 0.173i)T \) |
| 83 | \( 1 + (-0.642 - 0.766i)T \) |
| 89 | \( 1 + (0.984 - 0.173i)T \) |
| 97 | \( 1 + (0.5 - 0.866i)T \) |
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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
−20.51132111922333498290806990716, −20.12252865472542429082808141450, −19.317058649703958013140425953824, −18.2403376466720315827569388871, −17.679155175105344723342816294323, −16.64948254407247860117329502162, −16.2500957897714354295013167778, −15.35952418500073720817260790106, −14.957886945909436778061943489433, −13.84638339780926957307488744241, −13.059418776059549802467035108578, −12.0790541678089753250898535626, −11.6032957274196520894161528034, −10.44901232886793136139287124425, −9.92374950489175801006899996805, −9.27284103354954309346494425683, −8.52764146884903872021909454772, −7.10384562896410182092049495178, −6.65618768673664775783127736025, −5.54337154155629181402307590285, −4.72118847204086200607297943288, −4.142926081166874346168470009296, −2.89828679885030068503186682982, −2.33129387692638129814244385435, −0.392994582523189109482376690651,
0.70579309738428354500340588536, 1.938010448736372812237147305380, 2.90701369093764546288360006922, 3.6736265832457869408417139415, 5.09154269796198412231218770281, 5.84331630802607798416939866603, 6.49410163812652383189699887844, 7.36820770920556284328239349366, 8.104694354438654269277859514739, 8.88445485846697100275906110793, 10.15276619318987606197371753492, 10.53707824482037308803063445383, 11.72958310827930698562617949001, 12.27256039098928063872504544810, 13.14526484730150768317064457105, 13.55571632155444808209081059203, 14.42258284673187425115011850282, 15.55961276556155146864037185191, 16.18906599818938148128293794195, 17.01774551391474854290807663199, 17.666053068917220646879632162089, 18.4365515743237795565070149455, 19.27473150492634713839587131762, 19.61634937555518185891947693547, 20.43382083633614823997855676743