L(s) = 1 | + (−0.207 + 0.978i)2-s + (−0.913 − 0.406i)4-s + (−0.866 − 0.5i)7-s + (0.587 − 0.809i)8-s + (0.978 + 0.207i)11-s + (0.207 + 0.978i)13-s + (0.669 − 0.743i)14-s + (0.669 + 0.743i)16-s + (−0.587 + 0.809i)17-s + (0.809 + 0.587i)19-s + (−0.406 + 0.913i)22-s + (0.743 + 0.669i)23-s − 26-s + (0.587 + 0.809i)28-s + (−0.104 − 0.994i)29-s + ⋯ |
L(s) = 1 | + (−0.207 + 0.978i)2-s + (−0.913 − 0.406i)4-s + (−0.866 − 0.5i)7-s + (0.587 − 0.809i)8-s + (0.978 + 0.207i)11-s + (0.207 + 0.978i)13-s + (0.669 − 0.743i)14-s + (0.669 + 0.743i)16-s + (−0.587 + 0.809i)17-s + (0.809 + 0.587i)19-s + (−0.406 + 0.913i)22-s + (0.743 + 0.669i)23-s − 26-s + (0.587 + 0.809i)28-s + (−0.104 − 0.994i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.152 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.152 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5805480182 + 0.6773369950i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5805480182 + 0.6773369950i\) |
\(L(1)\) |
\(\approx\) |
\(0.7427610332 + 0.4306596532i\) |
\(L(1)\) |
\(\approx\) |
\(0.7427610332 + 0.4306596532i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-0.207 + 0.978i)T \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
| 11 | \( 1 + (0.978 + 0.207i)T \) |
| 13 | \( 1 + (0.207 + 0.978i)T \) |
| 17 | \( 1 + (-0.587 + 0.809i)T \) |
| 19 | \( 1 + (0.809 + 0.587i)T \) |
| 23 | \( 1 + (0.743 + 0.669i)T \) |
| 29 | \( 1 + (-0.104 - 0.994i)T \) |
| 31 | \( 1 + (-0.104 + 0.994i)T \) |
| 37 | \( 1 + (0.951 + 0.309i)T \) |
| 41 | \( 1 + (0.978 - 0.207i)T \) |
| 43 | \( 1 + (0.866 + 0.5i)T \) |
| 47 | \( 1 + (-0.994 + 0.104i)T \) |
| 53 | \( 1 + (-0.587 - 0.809i)T \) |
| 59 | \( 1 + (-0.978 + 0.207i)T \) |
| 61 | \( 1 + (-0.978 - 0.207i)T \) |
| 67 | \( 1 + (-0.994 - 0.104i)T \) |
| 71 | \( 1 + (0.809 - 0.587i)T \) |
| 73 | \( 1 + (0.951 - 0.309i)T \) |
| 79 | \( 1 + (0.104 + 0.994i)T \) |
| 83 | \( 1 + (0.406 + 0.913i)T \) |
| 89 | \( 1 + (0.309 + 0.951i)T \) |
| 97 | \( 1 + (0.994 - 0.104i)T \) |
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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
−26.37268128158501383909702270398, −25.36767574241085795620473895777, −24.449302414821620089948630324911, −22.816570997371032000954474304963, −22.46365903298824020047801910833, −21.583624613050628394105074719790, −20.28912079188145034271360272672, −19.80356958439861071684190763932, −18.723194366332730756251976087546, −17.9797152275590549957476057237, −16.87401162381319117032926088228, −15.82568555800332531671473741458, −14.54195067690820194463186862631, −13.37232399471304333460135099980, −12.638576710676543398345357429077, −11.61279397820728315494222266719, −10.71741002101525767222128345127, −9.40084615961974156085101597628, −8.99319222633624705714121640178, −7.5110055782194736956960511704, −6.097660134001581595237723517753, −4.78105940601273917476192591821, −3.41611134631763690699483329948, −2.581741465053474539489221745684, −0.833233521894645178669906503592,
1.35401607616184432282186002599, 3.587802546418556190514913925829, 4.50640219771536582231251343393, 6.0770463075622464569289550383, 6.72382258141224733567795197000, 7.78257803572970128611806102385, 9.15614394032867867283896468096, 9.676743397581719690754689134394, 11.04403039424980356816273769239, 12.47888470208209734918811306717, 13.544770574334838361587318841829, 14.31477147775614026888740357860, 15.39915657212014120879678739489, 16.38999882681447344324376082405, 17.01276621333804485734523577681, 18.01068189861081742957969539798, 19.25867816954373148325301241853, 19.68839437753330740458917496230, 21.28956975917728477654301109913, 22.4059010742107146524071142878, 23.029542620422784703468415677415, 24.007634594715084727092036308853, 24.90672504224214115591709482729, 25.7839763610673974556172307447, 26.54391268067715958374417201235