L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.254 − 0.254i)3-s + 1.00i·4-s + (−1.36 + 1.77i)5-s − 0.360·6-s + (−0.812 + 0.812i)7-s + (0.707 − 0.707i)8-s + 2.87i·9-s + (2.21 − 0.292i)10-s + 3.05i·11-s + (0.254 + 0.254i)12-s + (0.697 − 0.697i)13-s + 1.14·14-s + (0.105 + 0.798i)15-s − 1.00·16-s + (−2.89 + 2.89i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (0.147 − 0.147i)3-s + 0.500i·4-s + (−0.608 + 0.793i)5-s − 0.147·6-s + (−0.307 + 0.307i)7-s + (0.250 − 0.250i)8-s + 0.956i·9-s + (0.701 − 0.0926i)10-s + 0.920i·11-s + (0.0735 + 0.0735i)12-s + (0.193 − 0.193i)13-s + 0.307·14-s + (0.0272 + 0.206i)15-s − 0.250·16-s + (−0.702 + 0.702i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.660523 + 0.395571i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.660523 + 0.395571i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (1.36 - 1.77i)T \) |
| 23 | \( 1 + (-4.42 + 1.84i)T \) |
good | 3 | \( 1 + (-0.254 + 0.254i)T - 3iT^{2} \) |
| 7 | \( 1 + (0.812 - 0.812i)T - 7iT^{2} \) |
| 11 | \( 1 - 3.05iT - 11T^{2} \) |
| 13 | \( 1 + (-0.697 + 0.697i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.89 - 2.89i)T - 17iT^{2} \) |
| 19 | \( 1 - 2.26T + 19T^{2} \) |
| 29 | \( 1 - 6.96iT - 29T^{2} \) |
| 31 | \( 1 + 0.938T + 31T^{2} \) |
| 37 | \( 1 + (-4.39 + 4.39i)T - 37iT^{2} \) |
| 41 | \( 1 + 3.69T + 41T^{2} \) |
| 43 | \( 1 + (2.58 + 2.58i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.923 - 0.923i)T + 47iT^{2} \) |
| 53 | \( 1 + (8.88 + 8.88i)T + 53iT^{2} \) |
| 59 | \( 1 - 4.22iT - 59T^{2} \) |
| 61 | \( 1 + 10.6iT - 61T^{2} \) |
| 67 | \( 1 + (-2.79 + 2.79i)T - 67iT^{2} \) |
| 71 | \( 1 - 4.55T + 71T^{2} \) |
| 73 | \( 1 + (-0.803 + 0.803i)T - 73iT^{2} \) |
| 79 | \( 1 - 16.8T + 79T^{2} \) |
| 83 | \( 1 + (-8.37 - 8.37i)T + 83iT^{2} \) |
| 89 | \( 1 + 9.61T + 89T^{2} \) |
| 97 | \( 1 + (4.17 - 4.17i)T - 97iT^{2} \) |
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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
−12.35965690353013778308394676250, −11.11196442207186585705965321077, −10.67164357465350842604278852859, −9.548026753815751417938564351132, −8.419588298188695251048721222245, −7.50775795384118173053623484515, −6.62510476568280255263832816613, −4.85123944846451977777625046424, −3.39182563199797264271906494594, −2.15009366343650155685819595823,
0.75316518491385105288339773674, 3.37954686723261163161946570468, 4.67122298355005740948187457568, 6.02362958938765197885902734599, 7.09912561803964145157544444510, 8.219303428657322268086875176728, 9.061904829453078275325815311122, 9.726175052843590790472205581892, 11.18076443150765876337871603656, 11.83956352489379633753984835606