L(s) = 1 | + (0.836 − 1.14i)2-s + (1.51 + 2.62i)3-s + (−0.601 − 1.90i)4-s + (0.866 + 0.5i)5-s + (4.25 + 0.465i)6-s + (−2.57 − 0.602i)7-s + (−2.67 − 0.908i)8-s + (−3.08 + 5.33i)9-s + (1.29 − 0.569i)10-s + (1.03 − 0.598i)11-s + (4.08 − 4.46i)12-s − 4.83i·13-s + (−2.84 + 2.43i)14-s + 3.02i·15-s + (−3.27 + 2.29i)16-s + (2.20 − 1.27i)17-s + ⋯ |
L(s) = 1 | + (0.591 − 0.806i)2-s + (0.873 + 1.51i)3-s + (−0.300 − 0.953i)4-s + (0.387 + 0.223i)5-s + (1.73 + 0.190i)6-s + (−0.973 − 0.227i)7-s + (−0.946 − 0.321i)8-s + (−1.02 + 1.77i)9-s + (0.409 − 0.180i)10-s + (0.312 − 0.180i)11-s + (1.18 − 1.28i)12-s − 1.34i·13-s + (−0.759 + 0.650i)14-s + 0.781i·15-s + (−0.819 + 0.573i)16-s + (0.534 − 0.308i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0510i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0510i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.68904 - 0.0431474i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.68904 - 0.0431474i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.836 + 1.14i)T \) |
| 5 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 + (2.57 + 0.602i)T \) |
good | 3 | \( 1 + (-1.51 - 2.62i)T + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-1.03 + 0.598i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 4.83iT - 13T^{2} \) |
| 17 | \( 1 + (-2.20 + 1.27i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.711 - 1.23i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.02 + 2.90i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 0.774T + 29T^{2} \) |
| 31 | \( 1 + (-3.31 - 5.74i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.55 - 4.42i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 7.46iT - 41T^{2} \) |
| 43 | \( 1 - 1.38iT - 43T^{2} \) |
| 47 | \( 1 + (-0.535 + 0.927i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.68 + 2.91i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.94 + 8.55i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-8.31 - 4.79i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (9.14 - 5.27i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 16.3iT - 71T^{2} \) |
| 73 | \( 1 + (-0.0927 + 0.0535i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-9.32 - 5.38i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 + (3.41 + 1.97i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 8.71iT - 97T^{2} \) |
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
−13.35280209327292798397472976634, −12.19917434054029476535792882017, −10.68076238876206467623859346615, −10.13169854678544455357345304289, −9.528011266322131514386731159238, −8.328498344095654982623586675796, −6.18172008516647596428480056046, −4.91852848930117050261153234535, −3.58780630399374764527063579619, −2.87387437052319920157967322820,
2.27682159207375784420897951913, 3.81693715062581311786679163688, 5.93092431674634112284115886856, 6.68955398333934368313667063265, 7.59220977332708715184127812881, 8.754876362646814371150583711645, 9.491288617156262209965605468748, 11.93894575254801000486065122150, 12.42720423433202367646731669717, 13.46139005442659148763443990847