L(s) = 1 | + (0.133 + 0.5i)2-s − 1.73·3-s + (1.5 − 0.866i)4-s + (2 + i)5-s + (−0.232 − 0.866i)6-s + (−2.5 + 0.866i)7-s + (1.36 + 1.36i)8-s + 2.99·9-s + (−0.232 + 1.13i)10-s + 1.46·11-s + (−2.59 + 1.49i)12-s + (−0.267 − i)13-s + (−0.767 − 1.13i)14-s + (−3.46 − 1.73i)15-s + (1.23 − 2.13i)16-s + (2.73 − 0.732i)17-s + ⋯ |
L(s) = 1 | + (0.0947 + 0.353i)2-s − 1.00·3-s + (0.750 − 0.433i)4-s + (0.894 + 0.447i)5-s + (−0.0947 − 0.353i)6-s + (−0.944 + 0.327i)7-s + (0.482 + 0.482i)8-s + 0.999·9-s + (−0.0733 + 0.358i)10-s + 0.441·11-s + (−0.749 + 0.433i)12-s + (−0.0743 − 0.277i)13-s + (−0.205 − 0.303i)14-s + (−0.894 − 0.447i)15-s + (0.308 − 0.533i)16-s + (0.662 − 0.177i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.28727 + 0.444330i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28727 + 0.444330i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 1.73T \) |
| 5 | \( 1 + (-2 - i)T \) |
| 7 | \( 1 + (2.5 - 0.866i)T \) |
good | 2 | \( 1 + (-0.133 - 0.5i)T + (-1.73 + i)T^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 + (0.267 + i)T + (-11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + (-2.73 + 0.732i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-3.36 - 5.83i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.63 - 1.63i)T + 23iT^{2} \) |
| 29 | \( 1 + (-6 + 3.46i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (8.83 - 5.09i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.09 + 1.09i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (8.19 + 4.73i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.5 + 5.59i)T + (-37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (-8.59 + 2.30i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (3.36 - 0.901i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (1.73 + 3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (8.13 + 4.69i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (8.33 + 2.23i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + (0.169 + 0.633i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (9.92 + 5.73i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.36 + 0.366i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (0.598 + 1.03i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.830 + 3.09i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−11.88463705644405079240898094435, −10.55175675541670438460041123634, −10.23292787992998466212253863988, −9.257311323211612110919908467729, −7.41586425351249769546076782423, −6.70259610839748692798263668379, −5.81640516647960740647113602964, −5.36587030356323068513949997206, −3.30990931373935617882119816718, −1.63071287078507208076661067811,
1.30546397913012155492974109565, 3.00536102888674295639694855635, 4.45915260506102175926438040120, 5.74192982056113064527463401039, 6.64661266107380402057013430666, 7.30778070142123944497219917324, 9.052374576806589374346117886438, 9.941102689190855926543496268761, 10.68319457694466879736186173687, 11.62240056203537346094567777684