L(s) = 1 | + (−0.155 − 0.269i)2-s + (−0.514 − 0.891i)3-s + (0.951 − 1.64i)4-s + (−0.160 + 0.277i)6-s − 3.28·7-s − 1.21·8-s + (0.969 − 1.67i)9-s − 5.16·11-s − 1.95·12-s + (−1.76 + 3.06i)13-s + (0.510 + 0.883i)14-s + (−1.71 − 2.96i)16-s + (0.504 + 0.874i)17-s − 0.603·18-s + (2.42 + 3.62i)19-s + ⋯ |
L(s) = 1 | + (−0.109 − 0.190i)2-s + (−0.297 − 0.514i)3-s + (0.475 − 0.824i)4-s + (−0.0653 + 0.113i)6-s − 1.23·7-s − 0.429·8-s + (0.323 − 0.559i)9-s − 1.55·11-s − 0.565·12-s + (−0.490 + 0.849i)13-s + (0.136 + 0.236i)14-s + (−0.428 − 0.742i)16-s + (0.122 + 0.211i)17-s − 0.142·18-s + (0.555 + 0.831i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.146i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0418423 + 0.568603i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0418423 + 0.568603i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (-2.42 - 3.62i)T \) |
good | 2 | \( 1 + (0.155 + 0.269i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (0.514 + 0.891i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + 3.28T + 7T^{2} \) |
| 11 | \( 1 + 5.16T + 11T^{2} \) |
| 13 | \( 1 + (1.76 - 3.06i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.504 - 0.874i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-3.83 + 6.63i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.01 + 3.48i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4.60T + 31T^{2} \) |
| 37 | \( 1 + 6.48T + 37T^{2} \) |
| 41 | \( 1 + (-3.40 - 5.89i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.15 + 5.46i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.92 + 3.32i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.55 + 6.16i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6.73 + 11.6i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.06 - 5.31i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.59 + 9.69i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.227 - 0.394i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.06 + 3.57i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.44 + 2.50i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 5.50T + 83T^{2} \) |
| 89 | \( 1 + (-3.56 + 6.17i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.41 - 9.37i)T + (-48.5 + 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.32809884221326706579902005774, −9.939863288178888537540994607591, −8.972735366119844676683989734950, −7.52413151649876329748431709850, −6.70663990490204204668848719098, −6.05934850837609855269038378923, −4.99020000725626619349971781804, −3.30063317785917401012725033145, −2.07592140050854673686409899301, −0.34063095852438283157953935224,
2.67928445108043370135981643861, 3.39235760829702096538997892795, 4.96475896142914929725905770808, 5.76155718095193681174931258220, 7.33084391420842670678441288904, 7.43502554301780282038583810703, 8.825565329562985984239285844326, 9.807825318758676312048628202564, 10.56113285119488086439643186392, 11.29533497100677146441135275292