L(s) = 1 | + (−0.407 − 0.913i)2-s + (0.560 − 0.828i)3-s + (−0.668 + 0.744i)4-s + (−0.844 − 0.536i)5-s + (−0.984 − 0.174i)6-s + (−0.977 − 0.212i)7-s + (0.951 + 0.307i)8-s + (−0.371 − 0.928i)9-s + (−0.145 + 0.989i)10-s + (0.987 − 0.155i)11-s + (0.241 + 0.970i)12-s + (0.987 + 0.155i)13-s + (0.203 + 0.979i)14-s + (−0.917 + 0.398i)15-s + (−0.107 − 0.994i)16-s + (0.592 − 0.805i)17-s + ⋯ |
L(s) = 1 | + (−0.407 − 0.913i)2-s + (0.560 − 0.828i)3-s + (−0.668 + 0.744i)4-s + (−0.844 − 0.536i)5-s + (−0.984 − 0.174i)6-s + (−0.977 − 0.212i)7-s + (0.951 + 0.307i)8-s + (−0.371 − 0.928i)9-s + (−0.145 + 0.989i)10-s + (0.987 − 0.155i)11-s + (0.241 + 0.970i)12-s + (0.987 + 0.155i)13-s + (0.203 + 0.979i)14-s + (−0.917 + 0.398i)15-s + (−0.107 − 0.994i)16-s + (0.592 − 0.805i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.05634566898 - 1.106414984i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.05634566898 - 1.106414984i\) |
\(L(1)\) |
\(\approx\) |
\(0.5387887446 - 0.7017465166i\) |
\(L(1)\) |
\(\approx\) |
\(0.5387887446 - 0.7017465166i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 967 | \( 1 \) |
good | 2 | \( 1 + (-0.407 - 0.913i)T \) |
| 3 | \( 1 + (0.560 - 0.828i)T \) |
| 5 | \( 1 + (-0.844 - 0.536i)T \) |
| 7 | \( 1 + (-0.977 - 0.212i)T \) |
| 11 | \( 1 + (0.987 - 0.155i)T \) |
| 13 | \( 1 + (0.987 + 0.155i)T \) |
| 17 | \( 1 + (0.592 - 0.805i)T \) |
| 19 | \( 1 + (0.996 - 0.0779i)T \) |
| 23 | \( 1 + (0.924 - 0.380i)T \) |
| 29 | \( 1 + (-0.957 + 0.288i)T \) |
| 31 | \( 1 + (0.0876 - 0.996i)T \) |
| 37 | \( 1 + (0.737 - 0.675i)T \) |
| 41 | \( 1 + (0.962 - 0.269i)T \) |
| 43 | \( 1 + (-0.184 - 0.982i)T \) |
| 47 | \( 1 + (-0.883 - 0.468i)T \) |
| 53 | \( 1 + (-0.576 - 0.816i)T \) |
| 59 | \( 1 + (0.279 + 0.960i)T \) |
| 61 | \( 1 + (0.962 - 0.269i)T \) |
| 67 | \( 1 + (0.0876 + 0.996i)T \) |
| 71 | \( 1 + (-0.407 + 0.913i)T \) |
| 73 | \( 1 + (-0.576 + 0.816i)T \) |
| 79 | \( 1 + (0.165 - 0.986i)T \) |
| 83 | \( 1 + (0.0487 + 0.998i)T \) |
| 89 | \( 1 + (0.0876 + 0.996i)T \) |
| 97 | \( 1 + (-0.900 - 0.433i)T \) |
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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
−22.46158401408863402099394391440, −21.55460581301499360471395722465, −20.3206495206321603438525876206, −19.55304155565982299265193312053, −19.141426520828548387616245026751, −18.33459650817486526846132931564, −17.18288078749383022892760640759, −16.26567497717686205742188905250, −15.96888320323989830440902626043, −15.04588103558436488308974950714, −14.640716724851297220518724591037, −13.71018866906353795887462942083, −12.790340037062642871880609725226, −11.446123636211995587555171075249, −10.68435912932016401370743762487, −9.70905304864170470613895608152, −9.20586756252644297004662087803, −8.27924083767066171313136832710, −7.55746884704202245341942114813, −6.55720231375585848139390684066, −5.81128171678071120647726930367, −4.602372148569757056365260377921, −3.64152090724047687186847171108, −3.16541625544235166258193771409, −1.29379306465890828858512885517,
0.656302128933533326890607303142, 1.264242273358246268263830355886, 2.68648810540182072155220660860, 3.566854968541553262839113204962, 3.99786057780301057503602854056, 5.55661257006864077210246572128, 6.90760831636815051606611887001, 7.488318364467019092587204429857, 8.49576539605423089828270188059, 9.15888645959438777764294019173, 9.70150444312890262981844865556, 11.241384090189360586147446868702, 11.65488866484122530039881220713, 12.562346971095694669815129563344, 13.134952282240240819664319104712, 13.836868743939277277960634382846, 14.81309792588469164429520036843, 16.16210733066528429588210363165, 16.58082739135157229022288484471, 17.62847265722665078239526978818, 18.687057718591718594174573623698, 18.99982562038642407723670777027, 19.73290530485800482997084562066, 20.45030970660990710138520762295, 20.79179188298647683471446566876