| L(s) = 1 | + (1.14 + 0.830i)2-s + (−0.0248 − 0.0847i)3-s + (0.621 + 1.90i)4-s + (−1.32 + 2.06i)5-s + (0.0418 − 0.117i)6-s + (−0.647 − 4.50i)7-s + (−0.865 + 2.69i)8-s + (2.51 − 1.61i)9-s + (−3.23 + 1.26i)10-s + (−0.477 + 1.04i)11-s + (0.145 − 0.0999i)12-s + (0.686 − 4.77i)13-s + (2.99 − 5.69i)14-s + (0.208 + 0.0611i)15-s + (−3.22 + 2.36i)16-s + (−2.34 + 2.03i)17-s + ⋯ |
| L(s) = 1 | + (0.809 + 0.586i)2-s + (−0.0143 − 0.0489i)3-s + (0.310 + 0.950i)4-s + (−0.594 + 0.924i)5-s + (0.0170 − 0.0480i)6-s + (−0.244 − 1.70i)7-s + (−0.306 + 0.951i)8-s + (0.839 − 0.539i)9-s + (−1.02 + 0.399i)10-s + (−0.144 + 0.315i)11-s + (0.0420 − 0.0288i)12-s + (0.190 − 1.32i)13-s + (0.801 − 1.52i)14-s + (0.0537 + 0.0157i)15-s + (−0.806 + 0.591i)16-s + (−0.569 + 0.493i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 - 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.623 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.21155 + 0.583918i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21155 + 0.583918i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.14 - 0.830i)T \) |
| 23 | \( 1 + (-2.00 + 4.35i)T \) |
| good | 3 | \( 1 + (0.0248 + 0.0847i)T + (-2.52 + 1.62i)T^{2} \) |
| 5 | \( 1 + (1.32 - 2.06i)T + (-2.07 - 4.54i)T^{2} \) |
| 7 | \( 1 + (0.647 + 4.50i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (0.477 - 1.04i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.686 + 4.77i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (2.34 - 2.03i)T + (2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (4.57 - 5.27i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (-0.809 - 0.934i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (0.611 - 2.08i)T + (-26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-1.13 - 1.76i)T + (-15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (1.45 + 0.937i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-1.97 + 0.578i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 - 8.32iT - 47T^{2} \) |
| 53 | \( 1 + (-0.322 + 0.0464i)T + (50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-5.07 - 0.730i)T + (56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-2.66 + 9.07i)T + (-51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-2.78 - 6.09i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (1.45 - 0.663i)T + (46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-2.38 + 2.75i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (-0.774 + 5.38i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-3.34 + 2.15i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (2.20 + 7.50i)T + (-74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (0.772 - 1.20i)T + (-40.2 - 88.2i)T^{2} \) |
| 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
−14.40702145240037234050440405854, −13.11531542495242516488648688060, −12.55881718023319049808956888252, −10.88847416304262653711562548782, −10.31286441760135226011677586580, −8.090076720358971355399272341832, −7.18029446485271097771651512451, −6.41138801803906228999363086380, −4.32865237832904758823152003567, −3.44174039805917484702738599173,
2.26101115351629662376998210342, 4.29158992518998996451451783830, 5.26686916973626460989585716452, 6.74037719191339199653479305395, 8.697331014117982499615012384942, 9.455387229300003226278081356439, 11.16354894927142264315821887241, 11.89251018508664892926931795693, 12.81813550611695868185251295481, 13.54679911234859327530392380309
