L(s) = 1 | + (0.885 − 0.464i)2-s + (−0.354 − 0.935i)3-s + (0.568 − 0.822i)4-s + (−0.748 + 0.663i)5-s + (−0.748 − 0.663i)6-s + (0.885 − 0.464i)7-s + (0.120 − 0.992i)8-s + (−0.748 + 0.663i)9-s + (−0.354 + 0.935i)10-s + (−0.970 − 0.239i)11-s + (−0.970 − 0.239i)12-s + (0.568 + 0.822i)13-s + (0.568 − 0.822i)14-s + (0.885 + 0.464i)15-s + (−0.354 − 0.935i)16-s + (0.120 + 0.992i)17-s + ⋯ |
L(s) = 1 | + (0.885 − 0.464i)2-s + (−0.354 − 0.935i)3-s + (0.568 − 0.822i)4-s + (−0.748 + 0.663i)5-s + (−0.748 − 0.663i)6-s + (0.885 − 0.464i)7-s + (0.120 − 0.992i)8-s + (−0.748 + 0.663i)9-s + (−0.354 + 0.935i)10-s + (−0.970 − 0.239i)11-s + (−0.970 − 0.239i)12-s + (0.568 + 0.822i)13-s + (0.568 − 0.822i)14-s + (0.885 + 0.464i)15-s + (−0.354 − 0.935i)16-s + (0.120 + 0.992i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8991539179 - 0.7516505647i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8991539179 - 0.7516505647i\) |
\(L(1)\) |
\(\approx\) |
\(1.153237824 - 0.6315225704i\) |
\(L(1)\) |
\(\approx\) |
\(1.153237824 - 0.6315225704i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 53 | \( 1 \) |
good | 2 | \( 1 + (0.885 - 0.464i)T \) |
| 3 | \( 1 + (-0.354 - 0.935i)T \) |
| 5 | \( 1 + (-0.748 + 0.663i)T \) |
| 7 | \( 1 + (0.885 - 0.464i)T \) |
| 11 | \( 1 + (-0.970 - 0.239i)T \) |
| 13 | \( 1 + (0.568 + 0.822i)T \) |
| 17 | \( 1 + (0.120 + 0.992i)T \) |
| 19 | \( 1 + (0.568 + 0.822i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (-0.970 + 0.239i)T \) |
| 31 | \( 1 + (-0.970 + 0.239i)T \) |
| 37 | \( 1 + (-0.354 - 0.935i)T \) |
| 41 | \( 1 + (-0.970 - 0.239i)T \) |
| 43 | \( 1 + (-0.354 + 0.935i)T \) |
| 47 | \( 1 + (-0.748 - 0.663i)T \) |
| 59 | \( 1 + (-0.748 - 0.663i)T \) |
| 61 | \( 1 + (0.120 - 0.992i)T \) |
| 67 | \( 1 + (0.568 - 0.822i)T \) |
| 71 | \( 1 + (-0.354 + 0.935i)T \) |
| 73 | \( 1 + (0.120 + 0.992i)T \) |
| 79 | \( 1 + (0.885 + 0.464i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (0.120 + 0.992i)T \) |
| 97 | \( 1 + (-0.748 + 0.663i)T \) |
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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
−33.524731969932227897581387346695, −32.45213604308256362002948753240, −31.50885645447488701849448706857, −30.74885091990031145673573882677, −28.97638056897874814444026095070, −27.83277179609271710588145965445, −26.854789794182820477542232938533, −25.46077593834938552693863420444, −24.16046678248009008967112077589, −23.25189484033139896484496342283, −22.20544771834235915444963020779, −20.7781096961209658041377905570, −20.49068053387608520822963485081, −18.01962586385899698493361070822, −16.68795329119121734155699615901, −15.57261268118008174119063941796, −15.06745430378664071264126340837, −13.27809559751477426705062474895, −11.866467958882154068762946501327, −10.99692499260237177302015922835, −8.839424771609448107476722467377, −7.59228195473776454446572124729, −5.367701981075557879110313745197, −4.82575923266381257163241235511, −3.19585609717101304775405314232,
1.74843178072362645827298977380, 3.58099007613297975173827326242, 5.31122537245342078556903535063, 6.82400371244705737355255238959, 7.970527360774459251385225052178, 10.766674921847484342655280788456, 11.32750068631229954104585519717, 12.63430159832082004924201582767, 13.8718659874542467945251968192, 14.86807220457389626805229843823, 16.45780181588924157224815367338, 18.30607689513344378894021731438, 19.04655851272990681515040182669, 20.353536097801821141112546212573, 21.627207183169158233796252160942, 23.149196349989837935170575804698, 23.55569958603501131189734641708, 24.53439103121418307842367895877, 26.204637183478932011887658517952, 27.753670874626640612572133514855, 28.92593637751900800862731326435, 29.947149866591134511738299085635, 30.939617097607794907523517266339, 31.30714563507863312634150765854, 33.298815350066214275062024924136