L(s) = 1 | + (−0.581 − 0.813i)2-s + (−0.872 − 0.489i)3-s + (−0.322 + 0.946i)4-s + (−0.791 + 0.611i)5-s + (0.109 + 0.994i)6-s + (0.639 − 0.768i)7-s + (0.957 − 0.288i)8-s + (0.520 + 0.853i)9-s + (0.957 + 0.288i)10-s + (0.989 − 0.145i)11-s + (0.744 − 0.667i)12-s + (−0.581 − 0.813i)13-s + (−0.997 − 0.0729i)14-s + (0.989 − 0.145i)15-s + (−0.791 − 0.611i)16-s + (−0.976 − 0.217i)17-s + ⋯ |
L(s) = 1 | + (−0.581 − 0.813i)2-s + (−0.872 − 0.489i)3-s + (−0.322 + 0.946i)4-s + (−0.791 + 0.611i)5-s + (0.109 + 0.994i)6-s + (0.639 − 0.768i)7-s + (0.957 − 0.288i)8-s + (0.520 + 0.853i)9-s + (0.957 + 0.288i)10-s + (0.989 − 0.145i)11-s + (0.744 − 0.667i)12-s + (−0.581 − 0.813i)13-s + (−0.997 − 0.0729i)14-s + (0.989 − 0.145i)15-s + (−0.791 − 0.611i)16-s + (−0.976 − 0.217i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.09149853088 - 0.3951585016i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09149853088 - 0.3951585016i\) |
\(L(1)\) |
\(\approx\) |
\(0.4196087154 - 0.2943286346i\) |
\(L(1)\) |
\(\approx\) |
\(0.4196087154 - 0.2943286346i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 173 | \( 1 \) |
good | 2 | \( 1 + (-0.581 - 0.813i)T \) |
| 3 | \( 1 + (-0.872 - 0.489i)T \) |
| 5 | \( 1 + (-0.791 + 0.611i)T \) |
| 7 | \( 1 + (0.639 - 0.768i)T \) |
| 11 | \( 1 + (0.989 - 0.145i)T \) |
| 13 | \( 1 + (-0.581 - 0.813i)T \) |
| 17 | \( 1 + (-0.976 - 0.217i)T \) |
| 19 | \( 1 + (-0.997 + 0.0729i)T \) |
| 23 | \( 1 + (0.989 + 0.145i)T \) |
| 29 | \( 1 + (0.109 - 0.994i)T \) |
| 31 | \( 1 + (-0.872 + 0.489i)T \) |
| 37 | \( 1 + (-0.997 + 0.0729i)T \) |
| 41 | \( 1 + (0.639 - 0.768i)T \) |
| 43 | \( 1 + (-0.322 - 0.946i)T \) |
| 47 | \( 1 + (-0.457 - 0.889i)T \) |
| 53 | \( 1 + (0.905 - 0.424i)T \) |
| 59 | \( 1 + (-0.181 - 0.983i)T \) |
| 61 | \( 1 + (-0.976 + 0.217i)T \) |
| 67 | \( 1 + (-0.872 - 0.489i)T \) |
| 71 | \( 1 + (0.109 - 0.994i)T \) |
| 73 | \( 1 + (0.833 + 0.551i)T \) |
| 79 | \( 1 + (-0.457 + 0.889i)T \) |
| 83 | \( 1 + (0.252 + 0.967i)T \) |
| 89 | \( 1 + (-0.0365 - 0.999i)T \) |
| 97 | \( 1 + (0.252 - 0.967i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−27.6190342723280040240089599810, −27.304846946901805116245638083541, −26.172356703006772630190884439795, −24.71244081664994017072299438714, −24.21577850661069614769756381482, −23.31457613645124306882847745971, −22.30106853995136959221210463434, −21.2881554278631254212542546040, −19.896013325871979133919485127822, −18.98104908232071931155069953114, −17.81367010017930622214483038924, −16.97177942551838871296189353792, −16.29867819082498682848279530058, −15.16280799753969714996322651118, −14.683067664929864660388968480517, −12.73161856837683291696369243888, −11.62222659435367040110623370659, −10.86413181703933776132561658538, −9.20362090953509696326977398539, −8.79874044208386088821553742704, −7.24310667233507557377191735596, −6.231704040664670866210577036236, −4.89706686548221284425358448739, −4.32124343685219489079412284214, −1.50428185134913862133115462051,
0.477427065097297842446775491514, 2.0296205538923591538437925791, 3.71910440048114943832508516600, 4.77720543928152332521282530441, 6.7827643741108944558257065794, 7.47137718180476542449765098155, 8.61965813832657933025151136893, 10.33896980353468294260260383538, 10.995791168919183724969099559457, 11.7306263062549837953597863943, 12.71039921463739532940928507817, 13.89353310199424887956815693153, 15.31378343588537124397714478130, 16.8097602099040267605493882384, 17.37462406716755001687738236903, 18.275724663047346521954507602501, 19.3830135458423157564394789734, 19.88240981762867503023617142565, 21.281479228869463053783754959593, 22.429771802552605864386013629748, 22.90099991077787586823467043572, 24.08600119186744266607284655322, 25.1592656556109574073002595530, 26.64065446422233675499431550029, 27.34965472462228671381725175241