| L(s) = 1 | + (0.000942 − 4.94e−5i)2-s + (1.67 − 2.07i)3-s + (−1.98 + 0.209i)4-s + (2.16 − 0.564i)5-s + (0.00147 − 0.00203i)6-s + (−2.21 − 1.45i)7-s + (−0.00372 + 0.000590i)8-s + (−0.853 − 4.01i)9-s + (0.00201 − 0.000639i)10-s + (4.83 + 1.02i)11-s + (−2.90 + 4.47i)12-s + (−4.82 + 2.45i)13-s + (−0.00215 − 0.00126i)14-s + (2.45 − 5.43i)15-s + (3.91 − 0.831i)16-s + (0.276 − 0.720i)17-s + ⋯ |
| L(s) = 1 | + (0.000666 − 3.49e−5i)2-s + (0.968 − 1.19i)3-s + (−0.994 + 0.104i)4-s + (0.967 − 0.252i)5-s + (0.000603 − 0.000831i)6-s + (−0.835 − 0.549i)7-s + (−0.00131 + 0.000208i)8-s + (−0.284 − 1.33i)9-s + (0.000636 − 0.000202i)10-s + (1.45 + 0.309i)11-s + (−0.838 + 1.29i)12-s + (−1.33 + 0.681i)13-s + (−0.000576 − 0.000337i)14-s + (0.635 − 1.40i)15-s + (0.978 − 0.207i)16-s + (0.0671 − 0.174i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.321 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.321 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.11072 - 0.795517i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11072 - 0.795517i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.16 + 0.564i)T \) |
| 7 | \( 1 + (2.21 + 1.45i)T \) |
| good | 2 | \( 1 + (-0.000942 + 4.94e-5i)T + (1.98 - 0.209i)T^{2} \) |
| 3 | \( 1 + (-1.67 + 2.07i)T + (-0.623 - 2.93i)T^{2} \) |
| 11 | \( 1 + (-4.83 - 1.02i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (4.82 - 2.45i)T + (7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.276 + 0.720i)T + (-12.6 - 11.3i)T^{2} \) |
| 19 | \( 1 + (-0.101 + 0.962i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (-0.143 - 2.73i)T + (-22.8 + 2.40i)T^{2} \) |
| 29 | \( 1 + (-2.32 - 3.19i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-3.67 - 8.24i)T + (-20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (1.77 + 1.15i)T + (15.0 + 33.8i)T^{2} \) |
| 41 | \( 1 + (6.23 + 2.02i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (2.74 - 2.74i)T - 43iT^{2} \) |
| 47 | \( 1 + (4.18 - 1.60i)T + (34.9 - 31.4i)T^{2} \) |
| 53 | \( 1 + (0.179 + 0.145i)T + (11.0 + 51.8i)T^{2} \) |
| 59 | \( 1 + (-2.95 - 3.28i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (9.04 + 8.14i)T + (6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-0.672 - 0.258i)T + (49.7 + 44.8i)T^{2} \) |
| 71 | \( 1 + (-3.49 + 2.53i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.442 - 0.681i)T + (-29.6 + 66.6i)T^{2} \) |
| 79 | \( 1 + (-2.01 + 4.51i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (2.09 + 13.2i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (7.76 - 8.62i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (-2.13 + 13.4i)T + (-92.2 - 29.9i)T^{2} \) |
| 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
−12.65032967101002199228570278501, −12.17609185284501537289434622621, −9.992740885701981033355439686451, −9.347857519982908611151133730648, −8.651829798671872737714679875864, −7.20961152024508473874024893795, −6.53766898517573956048095796399, −4.76278991691560520784956229798, −3.19766433205701832340081255744, −1.48641406678540100449583055191,
2.75663798928704502256733736974, 3.92241820065715560134713296203, 5.15946914647570019248208228475, 6.38305369686292704477704362347, 8.310885016782267738125743811074, 9.204002453298126265220466463200, 9.768125700263175605949999152946, 10.23335210619195075195767113533, 12.07854621530704978606393606269, 13.21235431491799340277910220522
