L(s) = 1 | + (−0.909 − 0.415i)3-s + (0.540 + 0.841i)7-s + (0.654 + 0.755i)9-s + (−0.142 + 0.989i)11-s + (0.540 − 0.841i)13-s + (0.281 − 0.959i)17-s + (0.959 − 0.281i)19-s + (−0.142 − 0.989i)21-s + (−0.281 − 0.959i)27-s + (−0.959 − 0.281i)29-s + (0.415 + 0.909i)31-s + (0.540 − 0.841i)33-s + (−0.755 + 0.654i)37-s + (−0.841 + 0.540i)39-s + (−0.654 + 0.755i)41-s + ⋯ |
L(s) = 1 | + (−0.909 − 0.415i)3-s + (0.540 + 0.841i)7-s + (0.654 + 0.755i)9-s + (−0.142 + 0.989i)11-s + (0.540 − 0.841i)13-s + (0.281 − 0.959i)17-s + (0.959 − 0.281i)19-s + (−0.142 − 0.989i)21-s + (−0.281 − 0.959i)27-s + (−0.959 − 0.281i)29-s + (0.415 + 0.909i)31-s + (0.540 − 0.841i)33-s + (−0.755 + 0.654i)37-s + (−0.841 + 0.540i)39-s + (−0.654 + 0.755i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.143549816 + 0.2443507903i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.143549816 + 0.2443507903i\) |
\(L(1)\) |
\(\approx\) |
\(0.9137035117 + 0.03259288635i\) |
\(L(1)\) |
\(\approx\) |
\(0.9137035117 + 0.03259288635i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (-0.909 - 0.415i)T \) |
| 7 | \( 1 + (0.540 + 0.841i)T \) |
| 11 | \( 1 + (-0.142 + 0.989i)T \) |
| 13 | \( 1 + (0.540 - 0.841i)T \) |
| 17 | \( 1 + (0.281 - 0.959i)T \) |
| 19 | \( 1 + (0.959 - 0.281i)T \) |
| 29 | \( 1 + (-0.959 - 0.281i)T \) |
| 31 | \( 1 + (0.415 + 0.909i)T \) |
| 37 | \( 1 + (-0.755 + 0.654i)T \) |
| 41 | \( 1 + (-0.654 + 0.755i)T \) |
| 43 | \( 1 + (0.909 + 0.415i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (0.540 + 0.841i)T \) |
| 59 | \( 1 + (0.841 + 0.540i)T \) |
| 61 | \( 1 + (0.415 + 0.909i)T \) |
| 67 | \( 1 + (-0.989 + 0.142i)T \) |
| 71 | \( 1 + (-0.142 - 0.989i)T \) |
| 73 | \( 1 + (-0.281 - 0.959i)T \) |
| 79 | \( 1 + (0.841 + 0.540i)T \) |
| 83 | \( 1 + (0.755 - 0.654i)T \) |
| 89 | \( 1 + (0.415 - 0.909i)T \) |
| 97 | \( 1 + (-0.755 - 0.654i)T \) |
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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
−21.915478485514667554993640357930, −20.85827714970371956138247895086, −20.74799303507720965566544521227, −19.29257440670225769104937967642, −18.64566954334812004196058835348, −17.70545937549167960160532628190, −17.00054694072163239031783596308, −16.38891294564996307385937707883, −15.713894992940596309563931807074, −14.5963614276754183187462190290, −13.8639478397018207077211866042, −12.997211628751706900750169746040, −11.913196527852927672616064876306, −11.19758312971703081132899790075, −10.68120993271486482290167130257, −9.78461333310932956160376684593, −8.78647331532088313579306423129, −7.76653869440218662972611365730, −6.840152889350214243538108656539, −5.89803072091552262123859708192, −5.21156137003763860110030026194, −4.01516560920486824812369674070, −3.59628382024407171688062227542, −1.755174999935870335744969865532, −0.7540964170329672408318724525,
1.01685548156210694709365246466, 2.03600387251441490939349510656, 3.130840336472634122596032218443, 4.6704416367603045663295704560, 5.25132784242300241588518322366, 5.997929641626199253939776690729, 7.14735108203469922825362255743, 7.73027361373566825802549949549, 8.82168689428293631929905982518, 9.870126913787832330244158635994, 10.65951841847245303000657290602, 11.73225061533792460583386900428, 12.00602209721055691164242370573, 13.01037075453080497461166999545, 13.74901752580998001208184089769, 14.954996895037066805955101654064, 15.60966005576044018290927967883, 16.37223823332500751752541075178, 17.41982943827310430117714174841, 18.14443218182620940919215877940, 18.325132383596953915722190932062, 19.471435813654485006743232396496, 20.5181799889083410211942032596, 21.10255237464070614403485704124, 22.25970712971443466269261783629