L(s) = 1 | + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + (0.207 + 0.978i)5-s + (−0.406 + 0.913i)7-s + (0.587 − 0.809i)8-s + (0.5 + 0.866i)10-s + (−0.104 + 0.994i)14-s + (0.309 − 0.951i)16-s + (−0.669 − 0.743i)17-s + (−0.406 − 0.913i)19-s + (0.743 + 0.669i)20-s + (0.5 − 0.866i)23-s + (−0.913 + 0.406i)25-s + (0.207 + 0.978i)28-s + (−0.809 + 0.587i)29-s + ⋯ |
L(s) = 1 | + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + (0.207 + 0.978i)5-s + (−0.406 + 0.913i)7-s + (0.587 − 0.809i)8-s + (0.5 + 0.866i)10-s + (−0.104 + 0.994i)14-s + (0.309 − 0.951i)16-s + (−0.669 − 0.743i)17-s + (−0.406 − 0.913i)19-s + (0.743 + 0.669i)20-s + (0.5 − 0.866i)23-s + (−0.913 + 0.406i)25-s + (0.207 + 0.978i)28-s + (−0.809 + 0.587i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4403578180 - 1.238386919i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4403578180 - 1.238386919i\) |
\(L(1)\) |
\(\approx\) |
\(1.531393837 - 0.1649277987i\) |
\(L(1)\) |
\(\approx\) |
\(1.531393837 - 0.1649277987i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 5 | \( 1 + (0.207 + 0.978i)T \) |
| 7 | \( 1 + (-0.406 + 0.913i)T \) |
| 17 | \( 1 + (-0.669 - 0.743i)T \) |
| 19 | \( 1 + (-0.406 - 0.913i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.809 + 0.587i)T \) |
| 31 | \( 1 + (-0.743 - 0.669i)T \) |
| 37 | \( 1 + (-0.406 + 0.913i)T \) |
| 41 | \( 1 + (-0.406 - 0.913i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 + (0.406 + 0.913i)T \) |
| 53 | \( 1 + (0.309 + 0.951i)T \) |
| 59 | \( 1 + (-0.587 - 0.809i)T \) |
| 61 | \( 1 + (-0.978 + 0.207i)T \) |
| 67 | \( 1 + (0.866 + 0.5i)T \) |
| 71 | \( 1 + (0.743 - 0.669i)T \) |
| 73 | \( 1 + (-0.587 - 0.809i)T \) |
| 79 | \( 1 + (-0.978 - 0.207i)T \) |
| 83 | \( 1 + (0.743 - 0.669i)T \) |
| 89 | \( 1 + (0.866 - 0.5i)T \) |
| 97 | \( 1 + (-0.743 - 0.669i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.2936840493496030266353173440, −20.342060374843560649376011147491, −19.96718543068139578639707376870, −19.10739946502117320738705713851, −17.6814359813456689084348266911, −17.01069265556165782617711354933, −16.55840456134295250816836847873, −15.7547150650062272920175555442, −14.96410853850695414262139587832, −14.025189550528626073249806228075, −13.33060042130837095894114955996, −12.83781537905087106953660623059, −12.11380319241506074968540782380, −11.09026462972125003150437176102, −10.34243681991883918375826911881, −9.27886881897342303560266791954, −8.339087163685864178404557050932, −7.52560125353723904918598704684, −6.67847042438437040183903209895, −5.7843322821833279553085695516, −5.07856856625057203637662367116, −3.98297724367599160419364423359, −3.689510995140644773810509725917, −2.17731546640356691927078537370, −1.31121380352423211114309725967,
0.15646142659307765119950644329, 1.820614733315650297542614826902, 2.65348898004288736936395778213, 3.13631213060625237661493109925, 4.315699234627546904609397079804, 5.23110263718100080205890862764, 6.10804092163911602649862121871, 6.729917015481270567865216513918, 7.47200104088705140502430597653, 8.94649578256684588113548587110, 9.61186557701340817087283332708, 10.73262072027585715725166031360, 11.14298394064640612822380817312, 12.034703382506816203631831556092, 12.855675877106013163672559561022, 13.53511435217322579207425398471, 14.3723451471308285555532129371, 15.1276228112184975459958716664, 15.53882856502253245596817984171, 16.4651634838746652219192733039, 17.56557956360623092605053555376, 18.65986389300924139044389650974, 18.84036543675557158927583310454, 19.883123715951773437309584353215, 20.62178496076174936674566083351