L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + (−0.5 − 0.866i)6-s − i·7-s − i·8-s + (0.5 + 0.866i)9-s + 11-s + i·12-s + (−0.866 + 0.5i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s − i·18-s + (0.5 − 0.866i)21-s + (−0.866 − 0.5i)22-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + (−0.5 − 0.866i)6-s − i·7-s − i·8-s + (0.5 + 0.866i)9-s + 11-s + i·12-s + (−0.866 + 0.5i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s − i·18-s + (0.5 − 0.866i)21-s + (−0.866 − 0.5i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9191528630 - 0.05884527032i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9191528630 - 0.05884527032i\) |
\(L(1)\) |
\(\approx\) |
\(0.9452109554 - 0.06355692387i\) |
\(L(1)\) |
\(\approx\) |
\(0.9452109554 - 0.06355692387i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + (-0.866 + 0.5i)T \) |
| 17 | \( 1 + (0.866 + 0.5i)T \) |
| 23 | \( 1 + (0.866 - 0.5i)T \) |
| 29 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 + (0.5 - 0.866i)T \) |
| 43 | \( 1 + (-0.866 - 0.5i)T \) |
| 47 | \( 1 + (-0.866 + 0.5i)T \) |
| 53 | \( 1 + (-0.866 + 0.5i)T \) |
| 59 | \( 1 + (-0.5 + 0.866i)T \) |
| 61 | \( 1 + (-0.5 - 0.866i)T \) |
| 67 | \( 1 + (0.866 - 0.5i)T \) |
| 71 | \( 1 + (0.5 - 0.866i)T \) |
| 73 | \( 1 + (-0.866 - 0.5i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + (-0.866 - 0.5i)T \) |
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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
−29.999784790118791622389068102405, −29.239006686510723176550355395463, −27.808525068609996749905408080657, −27.11678231071668760008881723454, −25.85314589376245965589644624706, −24.96570242834065587295872955955, −24.607917414055833160359005743663, −23.191186253814226835790871574621, −21.62961250893157715305055937495, −20.250114422530522082530394775336, −19.40359210175515103409499878287, −18.55868239202358882996257963073, −17.58513925508679577952902131993, −16.25958193128096865105953996142, −14.89008610455928670641290497733, −14.49111739253738591783252154434, −12.713539649326962386350214632900, −11.52789204258280998488223151785, −9.67845695439024174604692812303, −9.02653878264752100285167661727, −7.83223289911116261247481629873, −6.80383117295982513791911911156, −5.38199802318896379831780062290, −3.01352895224316356312594537840, −1.58348682469929495762951840284,
1.64524898387273256848166656262, 3.280817955322843498243209889044, 4.34338036253021673064181784620, 6.925612721765482170710184870257, 7.93085360561015145759414073561, 9.21934367625891441188778725846, 10.01392754016456614584385853345, 11.13206117608775653126856791853, 12.578807459511616275393282226606, 13.95123142725517641567263700722, 15.03658561919824872152800397615, 16.63811213004985408389921665450, 17.06079801642375459489721714729, 18.85181449382051277910892743847, 19.60280399375680085570424207946, 20.44026407351372806135985740228, 21.38171313516966552259827427514, 22.45812984064728908507401614377, 24.25721085818293273949101388628, 25.32732382418260580458770294862, 26.27184439362836933090795477601, 27.05535020422464662250435147172, 27.76103082598314813172572564066, 29.19933423769492854208036616345, 30.14478302374215218367829815622