L(s) = 1 | + (−1.55 + 2.56i)3-s + 5.60i·5-s + (−5.05 + 8.75i)7-s + (−4.18 − 7.96i)9-s + (−4.84 − 2.79i)11-s + (−9.41 + 16.3i)13-s + (−14.3 − 8.69i)15-s + (1.70 + 0.983i)17-s + (−12.6 − 14.1i)19-s + (−14.6 − 26.5i)21-s + (14.6 + 8.46i)23-s − 6.39·25-s + (26.9 + 1.60i)27-s + 12.2i·29-s + (3.88 + 6.73i)31-s + ⋯ |
L(s) = 1 | + (−0.517 + 0.855i)3-s + 1.12i·5-s + (−0.722 + 1.25i)7-s + (−0.465 − 0.885i)9-s + (−0.440 − 0.254i)11-s + (−0.724 + 1.25i)13-s + (−0.959 − 0.579i)15-s + (0.100 + 0.0578i)17-s + (−0.667 − 0.744i)19-s + (−0.697 − 1.26i)21-s + (0.637 + 0.367i)23-s − 0.255·25-s + (0.998 + 0.0593i)27-s + 0.423i·29-s + (0.125 + 0.217i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.311 + 0.950i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4980763811\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4980763811\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.55 - 2.56i)T \) |
| 19 | \( 1 + (12.6 + 14.1i)T \) |
good | 5 | \( 1 - 5.60iT - 25T^{2} \) |
| 7 | \( 1 + (5.05 - 8.75i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (4.84 + 2.79i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (9.41 - 16.3i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + (-1.70 - 0.983i)T + (144.5 + 250. i)T^{2} \) |
| 23 | \( 1 + (-14.6 - 8.46i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 12.2iT - 841T^{2} \) |
| 31 | \( 1 + (-3.88 - 6.73i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 47.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 36.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (20.2 + 35.1i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + 1.53iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-15.7 + 9.06i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + 25.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 63.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-12.0 + 20.9i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (106. + 61.3i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (10.0 - 17.3i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-5.16 - 8.94i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (39.0 + 22.5i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (141. - 81.7i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (90.0 + 155. i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.93922373541686272809797920398, −9.975427964513574189597963217122, −9.347310138597621986422279109392, −8.587052424828148798787539754764, −7.02433309231635150295227984416, −6.43969800318528638412182347810, −5.53509151091467494747062815390, −4.52623897613307637578470805032, −3.20770489442746380796753077891, −2.47794487601585765653881395939,
0.21678432147718878988383308735, 1.08671053154208834213633560193, 2.70391840376181631986955154912, 4.22918062434636720100437826166, 5.16647000468869747655366402152, 6.06573226128032936099556863390, 7.14678952049591678197095074467, 7.78886933866610973372849352425, 8.598451263614908189072997319514, 9.940410237646269575730871117293