L(s) = 1 | + (0.451 − 0.892i)2-s + (−0.782 − 0.622i)3-s + (−0.592 − 0.805i)4-s + (0.528 + 0.849i)5-s + (−0.909 + 0.416i)6-s + (0.180 + 0.983i)7-s + (−0.986 + 0.164i)8-s + (0.224 + 0.974i)9-s + (0.996 − 0.0880i)10-s + (−0.421 − 0.906i)11-s + (−0.0385 + 0.999i)12-s + (−0.480 − 0.876i)13-s + (0.959 + 0.282i)14-s + (0.115 − 0.993i)15-s + (−0.298 + 0.954i)16-s + (−0.319 − 0.947i)17-s + ⋯ |
L(s) = 1 | + (0.451 − 0.892i)2-s + (−0.782 − 0.622i)3-s + (−0.592 − 0.805i)4-s + (0.528 + 0.849i)5-s + (−0.909 + 0.416i)6-s + (0.180 + 0.983i)7-s + (−0.986 + 0.164i)8-s + (0.224 + 0.974i)9-s + (0.996 − 0.0880i)10-s + (−0.421 − 0.906i)11-s + (−0.0385 + 0.999i)12-s + (−0.480 − 0.876i)13-s + (0.959 + 0.282i)14-s + (0.115 − 0.993i)15-s + (−0.298 + 0.954i)16-s + (−0.319 − 0.947i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.257 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.257 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.005446797084 + 0.007084922512i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.005446797084 + 0.007084922512i\) |
\(L(1)\) |
\(\approx\) |
\(0.6406916921 - 0.3603286915i\) |
\(L(1)\) |
\(\approx\) |
\(0.6406916921 - 0.3603286915i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 571 | \( 1 \) |
good | 2 | \( 1 + (0.451 - 0.892i)T \) |
| 3 | \( 1 + (-0.782 - 0.622i)T \) |
| 5 | \( 1 + (0.528 + 0.849i)T \) |
| 7 | \( 1 + (0.180 + 0.983i)T \) |
| 11 | \( 1 + (-0.421 - 0.906i)T \) |
| 13 | \( 1 + (-0.480 - 0.876i)T \) |
| 17 | \( 1 + (-0.319 - 0.947i)T \) |
| 19 | \( 1 + (-0.834 + 0.551i)T \) |
| 23 | \( 1 + (-0.461 + 0.887i)T \) |
| 29 | \( 1 + (-0.821 + 0.569i)T \) |
| 31 | \( 1 + (-0.677 - 0.735i)T \) |
| 37 | \( 1 + (-0.982 - 0.186i)T \) |
| 41 | \( 1 + (-0.926 - 0.376i)T \) |
| 43 | \( 1 + (-0.857 - 0.514i)T \) |
| 47 | \( 1 + (0.975 - 0.218i)T \) |
| 53 | \( 1 + (-0.889 - 0.456i)T \) |
| 59 | \( 1 + (-0.879 + 0.475i)T \) |
| 61 | \( 1 + (-0.256 + 0.966i)T \) |
| 67 | \( 1 + (0.840 - 0.542i)T \) |
| 71 | \( 1 + (0.669 + 0.743i)T \) |
| 73 | \( 1 + (0.329 + 0.944i)T \) |
| 79 | \( 1 + (0.411 - 0.911i)T \) |
| 83 | \( 1 + (0.0935 - 0.995i)T \) |
| 89 | \( 1 + (0.815 + 0.578i)T \) |
| 97 | \( 1 + (0.00551 + 0.999i)T \) |
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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
−23.800420004252901742165007895307, −23.24418386717032523759018165782, −22.19060000893304506273793628001, −21.53171901009371009885271869921, −20.8042414817901582104854311840, −20.02489724396520939267819538294, −18.41635888210587457599041158647, −17.31274114223439101351742362065, −17.16486809344736143840167492753, −16.44670675232359449892948991085, −15.50000527070230124826081705895, −14.692932811221062529554342523779, −13.73542111251778737625110693235, −12.756710871651379357220976990948, −12.254527164036861247764209519239, −10.92069613969521855064564596700, −9.95861951209603997343897148646, −9.156986346353698682576512673883, −8.11626268676354262577795697262, −6.89401434838979096319855605739, −6.2819459479915841973489073347, −5.00721300686421495326204517722, −4.601753689149789881337841003531, −3.8210132975450978138427710969, −1.86400026900306759265896282028,
0.00405190954490111890255113132, 1.77309806212301486453526112070, 2.47828277911808543435564749298, 3.49011086929492811209326326179, 5.281772873832707704838655786044, 5.52956577503060940494089923695, 6.47728547489413974974698215562, 7.720959795931680777958616552108, 8.94900156098603221831601588498, 10.08740332526055866418122130829, 10.83379632663640438775524823123, 11.515512565416887201728129811675, 12.32425854720900905564084165126, 13.205516651224983296324523586075, 13.87500264488521129194249954445, 14.88771353916505857611213676659, 15.72424917959971724396614659893, 17.14074632937088347234464641361, 17.983821296795013025485399495981, 18.64955736834901512579069273208, 18.96505718831279186115314792401, 20.19701097259371202394806359793, 21.32731994312103666292399875413, 21.98555112967855007924224798228, 22.39257776954308119501867281298