L(s) = 1 | + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (0.309 − 0.951i)5-s + (−0.809 + 0.587i)6-s − 1.61·7-s + (−0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + (−0.809 − 0.587i)10-s + (0.190 − 0.587i)11-s + (0.309 + 0.951i)12-s + (−0.500 + 1.53i)14-s + (−0.809 + 0.587i)15-s + (0.309 + 0.951i)16-s + 18-s + ⋯ |
L(s) = 1 | + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (0.309 − 0.951i)5-s + (−0.809 + 0.587i)6-s − 1.61·7-s + (−0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + (−0.809 − 0.587i)10-s + (0.190 − 0.587i)11-s + (0.309 + 0.951i)12-s + (−0.500 + 1.53i)14-s + (−0.809 + 0.587i)15-s + (0.309 + 0.951i)16-s + 18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5365102457\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5365102457\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.309 + 0.951i)T \) |
| 3 | \( 1 + (0.809 + 0.587i)T \) |
| 5 | \( 1 + (-0.309 + 0.951i)T \) |
good | 7 | \( 1 + 1.61T + T^{2} \) |
| 11 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 29 | \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
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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
−10.40226539939400498846125965897, −9.692900452249329745331970838407, −8.992187543682593437721113946648, −7.79478188794710945819909703724, −6.22063280972913537765628957527, −5.94778937317783289954358582731, −4.76479160159235241548278419525, −3.62819773786799494361511988085, −2.20860206905284398658715132605, −0.61723419447780696887501190023,
3.10719051896162680976184441992, 3.88265714929399562876545015626, 5.17359885429417921345660262352, 6.12378930611367341329444425505, 6.68869211954584936440997489726, 7.31852904345565715371835095778, 8.950068246658059596287880910925, 9.723669460073739776267157193982, 10.20274901874850271417652662893, 11.31950005221788735452879341167