L(s) = 1 | + (0.842 + 1.13i)2-s + (−0.218 + 1.71i)3-s + (−0.581 + 1.91i)4-s + (−2.13 + 1.19i)6-s + 3.64·7-s + (−2.66 + 0.949i)8-s + (−2.90 − 0.750i)9-s + 5.07i·11-s + (−3.16 − 1.41i)12-s + 1.70·13-s + (3.06 + 4.14i)14-s + (−3.32 − 2.22i)16-s − 4.08·17-s + (−1.59 − 3.93i)18-s + 1.26·19-s + ⋯ |
L(s) = 1 | + (0.595 + 0.803i)2-s + (−0.126 + 0.992i)3-s + (−0.290 + 0.956i)4-s + (−0.872 + 0.489i)6-s + 1.37·7-s + (−0.941 + 0.335i)8-s + (−0.968 − 0.250i)9-s + 1.52i·11-s + (−0.912 − 0.409i)12-s + 0.473·13-s + (0.820 + 1.10i)14-s + (−0.830 − 0.556i)16-s − 0.989·17-s + (−0.375 − 0.926i)18-s + 0.290·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.235607 + 1.89325i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.235607 + 1.89325i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.842 - 1.13i)T \) |
| 3 | \( 1 + (0.218 - 1.71i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 3.64T + 7T^{2} \) |
| 11 | \( 1 - 5.07iT - 11T^{2} \) |
| 13 | \( 1 - 1.70T + 13T^{2} \) |
| 17 | \( 1 + 4.08T + 17T^{2} \) |
| 19 | \( 1 - 1.26T + 19T^{2} \) |
| 23 | \( 1 + 4.70iT - 23T^{2} \) |
| 29 | \( 1 + 1.06T + 29T^{2} \) |
| 31 | \( 1 - 4.86iT - 31T^{2} \) |
| 37 | \( 1 - 7.56T + 37T^{2} \) |
| 41 | \( 1 + 1.50iT - 41T^{2} \) |
| 43 | \( 1 + 3.43iT - 43T^{2} \) |
| 47 | \( 1 + 10.9iT - 47T^{2} \) |
| 53 | \( 1 - 8.87iT - 53T^{2} \) |
| 59 | \( 1 - 0.788iT - 59T^{2} \) |
| 61 | \( 1 - 0.627iT - 61T^{2} \) |
| 67 | \( 1 - 4.18iT - 67T^{2} \) |
| 71 | \( 1 - 6.21T + 71T^{2} \) |
| 73 | \( 1 - 4.21iT - 73T^{2} \) |
| 79 | \( 1 - 0.992iT - 79T^{2} \) |
| 83 | \( 1 - 7.72T + 83T^{2} \) |
| 89 | \( 1 - 11.5iT - 89T^{2} \) |
| 97 | \( 1 + 7.40iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.14669970474609071683079889919, −10.22639247822971483265082463291, −9.081825078249147466951913158599, −8.448544629183750854656932267922, −7.48218079169907086853778604587, −6.48304971966589660918948127965, −5.26804044053297538820062939501, −4.65142582409321561082516692383, −3.98421460625580394937876385019, −2.36203655454983665483009812218,
0.950393878809377982746422250319, 2.06561117371089591583693374102, 3.31029223337620757630400861080, 4.64182543309301725703755227744, 5.67937345764679966029395687514, 6.31709126376617966921241541330, 7.73146581065132421667222864299, 8.457489571621493823223950404651, 9.353915594806964632979769643171, 10.90416509884362004814922970498