L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (−0.500 + 0.866i)4-s + (1.23 − 1.86i)5-s − 0.999·6-s + (−2.59 + 0.5i)7-s − 3i·8-s + (−1 − 1.73i)9-s + (−0.133 + 2.23i)10-s + (−0.866 + 0.499i)12-s + 2i·13-s + (2 − 1.73i)14-s + (2 − i)15-s + (0.500 + 0.866i)16-s + (1.73 + i)17-s + (1.73 + i)18-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.499 + 0.288i)3-s + (−0.250 + 0.433i)4-s + (0.550 − 0.834i)5-s − 0.408·6-s + (−0.981 + 0.188i)7-s − 1.06i·8-s + (−0.333 − 0.577i)9-s + (−0.0423 + 0.705i)10-s + (−0.249 + 0.144i)12-s + 0.554i·13-s + (0.534 − 0.462i)14-s + (0.516 − 0.258i)15-s + (0.125 + 0.216i)16-s + (0.420 + 0.242i)17-s + (0.408 + 0.235i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.579207 + 0.176127i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.579207 + 0.176127i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.23 + 1.86i)T \) |
| 7 | \( 1 + (2.59 - 0.5i)T \) |
good | 2 | \( 1 + (0.866 - 0.5i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.866 - 0.5i)T + (1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + (-1.73 - i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.59 - 1.5i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 7T + 29T^{2} \) |
| 31 | \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.92 + 4i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 - 7iT - 43T^{2} \) |
| 47 | \( 1 + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.19 + 3i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5 + 8.66i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.33 + 2.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + (-5.19 - 3i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 11iT - 83T^{2} \) |
| 89 | \( 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 16iT - 97T^{2} \) |
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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
−16.59599889359784038047228436459, −16.01776057877680642951370267214, −14.38870768282604730026847817490, −13.08828761368256559470201926028, −12.16344503206688543355672839558, −9.663022480577508129781761027413, −9.301335416853110947527723441366, −7.965175190318869538144019382111, −6.08736534843365498823103052728, −3.70880498029801417858296829336,
2.70491994321594748957871473769, 5.73394084151602609198180591633, 7.47778755486552906944461559803, 9.152782447120841016587155346774, 10.12022840969341865944050524293, 11.19194163352309747159901154511, 13.23183529941576512737781814143, 13.99687654344275496005780677264, 15.13336516674055792988612381600, 16.75630795189748001635313477401