L(s) = 1 | + (−0.159 + 0.987i)2-s + (−0.949 − 0.314i)4-s + (0.993 + 0.110i)5-s + (0.461 − 0.887i)8-s + (−0.266 + 0.963i)10-s + (−0.904 + 0.426i)11-s + (−0.601 + 0.799i)13-s + (0.802 + 0.596i)16-s + (−0.234 + 0.972i)17-s + (−0.266 − 0.963i)19-s + (−0.909 − 0.416i)20-s + (−0.277 − 0.960i)22-s + (0.834 − 0.551i)23-s + (0.975 + 0.218i)25-s + (−0.693 − 0.720i)26-s + ⋯ |
L(s) = 1 | + (−0.159 + 0.987i)2-s + (−0.949 − 0.314i)4-s + (0.993 + 0.110i)5-s + (0.461 − 0.887i)8-s + (−0.266 + 0.963i)10-s + (−0.904 + 0.426i)11-s + (−0.601 + 0.799i)13-s + (0.802 + 0.596i)16-s + (−0.234 + 0.972i)17-s + (−0.266 − 0.963i)19-s + (−0.909 − 0.416i)20-s + (−0.277 − 0.960i)22-s + (0.834 − 0.551i)23-s + (0.975 + 0.218i)25-s + (−0.693 − 0.720i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1729255106 + 1.235582455i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1729255106 + 1.235582455i\) |
\(L(1)\) |
\(\approx\) |
\(0.7729004247 + 0.5653283778i\) |
\(L(1)\) |
\(\approx\) |
\(0.7729004247 + 0.5653283778i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 191 | \( 1 \) |
good | 2 | \( 1 + (-0.159 + 0.987i)T \) |
| 5 | \( 1 + (0.993 + 0.110i)T \) |
| 11 | \( 1 + (-0.904 + 0.426i)T \) |
| 13 | \( 1 + (-0.601 + 0.799i)T \) |
| 17 | \( 1 + (-0.234 + 0.972i)T \) |
| 19 | \( 1 + (-0.266 - 0.963i)T \) |
| 23 | \( 1 + (0.834 - 0.551i)T \) |
| 29 | \( 1 + (-0.490 + 0.871i)T \) |
| 31 | \( 1 + (0.926 + 0.376i)T \) |
| 37 | \( 1 + (-0.926 + 0.376i)T \) |
| 41 | \( 1 + (0.546 + 0.837i)T \) |
| 43 | \( 1 + (0.997 - 0.0660i)T \) |
| 47 | \( 1 + (0.287 - 0.957i)T \) |
| 53 | \( 1 + (0.982 + 0.186i)T \) |
| 59 | \( 1 + (0.528 + 0.849i)T \) |
| 61 | \( 1 + (0.381 - 0.924i)T \) |
| 67 | \( 1 + (-0.942 + 0.335i)T \) |
| 71 | \( 1 + (0.627 - 0.778i)T \) |
| 73 | \( 1 + (0.795 + 0.605i)T \) |
| 79 | \( 1 + (-0.999 - 0.0110i)T \) |
| 83 | \( 1 + (0.894 - 0.446i)T \) |
| 89 | \( 1 + (-0.0385 - 0.999i)T \) |
| 97 | \( 1 + (-0.828 + 0.560i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.112620013262883995097397091369, −17.65621217038073140604015360351, −17.10137256246351823053800930968, −16.31462739773542652324675420097, −15.41123667310630848223616621856, −14.54423408962367068210450277547, −13.74393261640221508768656020875, −13.40807958538377233362499291806, −12.62980775464251974272877464876, −12.11603286178563162002219171611, −11.06549533389800010934671478926, −10.62166545748691955128725490473, −9.8442362986611871654708539825, −9.44577693994586702782797487268, −8.56872381289021845531337589969, −7.8656724068412526819891340345, −7.09460454669556100523623562676, −5.738434421337336871103903677518, −5.4609028293045505549703682138, −4.63559510477478085464912819947, −3.66557076034651308432036171096, −2.576343375640406475180900052813, −2.49967212244769520301587724916, −1.29878244037470596119232635328, −0.40888897539825016407464646508,
1.06380024534191975942299331642, 2.07132313053796324235950591430, 2.82464287981388332804241914069, 4.08306049320607722660553888553, 4.90488288770050413093514205324, 5.3018130327852800974994343203, 6.28021601929254473706873214258, 6.8385563931046892934607408340, 7.387614439462711079463867428699, 8.46488636645637534603539580741, 8.95750695565761270273409693414, 9.661275306656389181356523389188, 10.413876840071909363717584180288, 10.84838857119531262940051991303, 12.19081722005807272641544288880, 12.96194691823451163825351323417, 13.36157285489619266529332417706, 14.14150975468527375276648352750, 14.82289745156138237524032033166, 15.275331680625066662473627120772, 16.146416678485654258514704379748, 16.87360458188307104011515097549, 17.368689648233036773735958289, 17.89521632002056043803085533216, 18.61908223287273055310821644494