| L(s) = 1 | + (−1.13 + 2.22i)2-s + (−0.270 + 1.71i)3-s + (−1.30 − 1.79i)4-s + (−4.95 + 0.689i)5-s + (−3.49 − 2.53i)6-s + (3.10 + 3.10i)7-s + (−4.38 + 0.694i)8-s + (−2.85 − 0.927i)9-s + (4.07 − 11.7i)10-s + (−2.50 − 7.72i)11-s + (3.42 − 1.74i)12-s + (1.49 + 2.93i)13-s + (−10.4 + 3.38i)14-s + (0.161 − 8.65i)15-s + (6.16 − 18.9i)16-s + (4.41 + 27.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.566 + 1.11i)2-s + (−0.0903 + 0.570i)3-s + (−0.326 − 0.449i)4-s + (−0.990 + 0.137i)5-s + (−0.582 − 0.423i)6-s + (0.443 + 0.443i)7-s + (−0.547 + 0.0867i)8-s + (−0.317 − 0.103i)9-s + (0.407 − 1.17i)10-s + (−0.228 − 0.701i)11-s + (0.285 − 0.145i)12-s + (0.115 + 0.225i)13-s + (−0.743 + 0.241i)14-s + (0.0107 − 0.577i)15-s + (0.385 − 1.18i)16-s + (0.259 + 1.63i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.181i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.983 + 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0589115 - 0.643451i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0589115 - 0.643451i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.270 - 1.71i)T \) |
| 5 | \( 1 + (4.95 - 0.689i)T \) |
| good | 2 | \( 1 + (1.13 - 2.22i)T + (-2.35 - 3.23i)T^{2} \) |
| 7 | \( 1 + (-3.10 - 3.10i)T + 49iT^{2} \) |
| 11 | \( 1 + (2.50 + 7.72i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (-1.49 - 2.93i)T + (-99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (-4.41 - 27.8i)T + (-274. + 89.3i)T^{2} \) |
| 19 | \( 1 + (15.2 - 20.9i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (-17.4 - 8.88i)T + (310. + 427. i)T^{2} \) |
| 29 | \( 1 + (-20.3 - 28.0i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (-8.77 - 6.37i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (21.6 - 11.0i)T + (804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-13.9 + 42.8i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (8.53 - 8.53i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (39.8 + 6.30i)T + (2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (-14.2 + 90.1i)T + (-2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (-101. - 32.8i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (4.73 + 14.5i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-4.18 - 26.4i)T + (-4.26e3 + 1.38e3i)T^{2} \) |
| 71 | \( 1 + (41.3 - 30.0i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-101. - 51.7i)T + (3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-21.7 - 29.9i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (102. - 16.2i)T + (6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-108. + 35.2i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (58.1 + 9.20i)T + (8.94e3 + 2.90e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.02789159757814382471917563662, −14.52761024949098361317480646607, −12.54934810759739705225738334504, −11.46803679251073384578047081068, −10.33911987439997096061535600085, −8.525879853249576661676758207046, −8.313451300457341214722666409543, −6.73412484629724613625483239325, −5.43208311700514715157272811783, −3.61371721916994656457080590873,
0.68356858869797798509561909430, 2.73254816535200711701714354215, 4.66289809319015125973310636738, 6.88867410438141190370359743931, 8.043421679992417538502622032852, 9.284321181913136102642438163561, 10.67013997007811560275858275340, 11.48316616412233143290217823587, 12.25971839226017599423971964664, 13.32505842404223756481204312812
