L(s) = 1 | + (−0.156 + 0.987i)2-s + (−0.987 + 0.156i)3-s + (−0.951 − 0.309i)4-s − i·6-s + (−0.852 − 0.522i)7-s + (0.453 − 0.891i)8-s + (0.951 − 0.309i)9-s + (−0.522 + 0.852i)11-s + (0.987 + 0.156i)12-s + (0.233 + 0.972i)13-s + (0.649 − 0.760i)14-s + (0.809 + 0.587i)16-s + (0.852 − 0.522i)17-s + (0.156 + 0.987i)18-s + (0.760 − 0.649i)19-s + ⋯ |
L(s) = 1 | + (−0.156 + 0.987i)2-s + (−0.987 + 0.156i)3-s + (−0.951 − 0.309i)4-s − i·6-s + (−0.852 − 0.522i)7-s + (0.453 − 0.891i)8-s + (0.951 − 0.309i)9-s + (−0.522 + 0.852i)11-s + (0.987 + 0.156i)12-s + (0.233 + 0.972i)13-s + (0.649 − 0.760i)14-s + (0.809 + 0.587i)16-s + (0.852 − 0.522i)17-s + (0.156 + 0.987i)18-s + (0.760 − 0.649i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.0009257074593 + 0.001045613037i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0009257074593 + 0.001045613037i\) |
\(L(1)\) |
\(\approx\) |
\(0.4687017682 + 0.2573480648i\) |
\(L(1)\) |
\(\approx\) |
\(0.4687017682 + 0.2573480648i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 \) |
good | 2 | \( 1 + (-0.156 + 0.987i)T \) |
| 3 | \( 1 + (-0.987 + 0.156i)T \) |
| 7 | \( 1 + (-0.852 - 0.522i)T \) |
| 11 | \( 1 + (-0.522 + 0.852i)T \) |
| 13 | \( 1 + (0.233 + 0.972i)T \) |
| 17 | \( 1 + (0.852 - 0.522i)T \) |
| 19 | \( 1 + (0.760 - 0.649i)T \) |
| 23 | \( 1 + (-0.760 + 0.649i)T \) |
| 29 | \( 1 + (-0.156 + 0.987i)T \) |
| 31 | \( 1 + (-0.522 - 0.852i)T \) |
| 37 | \( 1 + (-0.996 + 0.0784i)T \) |
| 41 | \( 1 + (-0.707 - 0.707i)T \) |
| 43 | \( 1 + (-0.522 - 0.852i)T \) |
| 47 | \( 1 + (0.987 + 0.156i)T \) |
| 53 | \( 1 + (-0.156 + 0.987i)T \) |
| 59 | \( 1 + (0.891 - 0.453i)T \) |
| 61 | \( 1 + (-0.891 - 0.453i)T \) |
| 67 | \( 1 + (0.987 - 0.156i)T \) |
| 71 | \( 1 + (0.972 - 0.233i)T \) |
| 73 | \( 1 + (0.233 - 0.972i)T \) |
| 79 | \( 1 + (-0.707 - 0.707i)T \) |
| 83 | \( 1 + (-0.809 + 0.587i)T \) |
| 89 | \( 1 + (0.972 - 0.233i)T \) |
| 97 | \( 1 + (-0.809 + 0.587i)T \) |
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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
−17.31659283001704777220234936370, −16.653088223598079351725635212723, −16.11239359925210074172415724041, −15.488174820232876598234887080499, −14.44570439083187392736572981055, −13.62611272801258411663429396852, −13.017150740374376994809990451175, −12.49583652157494733830451063791, −11.99791330121016341568375494295, −11.33156620421130613298889891589, −10.54374428703458196836277863798, −10.09188272136406150796772205585, −9.64576515647078441685568102388, −8.411139832221153638086652948218, −8.14559779332302953785936774284, −7.14259048606606003166461509771, −6.09609986193287774275009083070, −5.595416942544678620360709176601, −5.1604425443215052877400867559, −3.961931152311539740170486115009, −3.38911099128340605759660588894, −2.69743684322578208340829992305, −1.686478753979375459657109881609, −0.85210040267956921920531358265, −0.00064905786288854641681025503,
0.98395807053571322977368375072, 1.93505576542571081369157709729, 3.457050299200510285022289107642, 3.93924928988503507100719689478, 4.88246581735075571718146340793, 5.32318698596547080627827125417, 6.06610819496551706159159224887, 6.8987470641108746037137341381, 7.185433108664887486069789472127, 7.80162738450117354684505931763, 9.141191264736951635115036181242, 9.44307658022367109311840879794, 10.180300065665707824401093862075, 10.64790171010367192304830298907, 11.71466293928216065422864703722, 12.31112629305297527219955440117, 12.99750944448512084197759342442, 13.72932254519195851623089434570, 14.17981343480481344363652418895, 15.30453990067798264082278221296, 15.74612620023348686689785301626, 16.18449492618513801261506127329, 16.97507576242803897591868456176, 17.180770254807965481101572860095, 18.18417068982463071723589958312