L(s) = 1 | + (0.814 − 0.580i)2-s + (−0.436 + 0.899i)3-s + (0.327 − 0.945i)4-s + (0.959 − 0.281i)5-s + (0.165 + 0.986i)6-s + (0.999 − 0.0237i)7-s + (−0.281 − 0.959i)8-s + (−0.618 − 0.786i)9-s + (0.618 − 0.786i)10-s + (−0.690 − 0.723i)11-s + (0.707 + 0.707i)12-s + (0.800 − 0.599i)14-s + (−0.165 + 0.986i)15-s + (−0.786 − 0.618i)16-s + (0.814 + 0.580i)17-s + (−0.959 − 0.281i)18-s + ⋯ |
L(s) = 1 | + (0.814 − 0.580i)2-s + (−0.436 + 0.899i)3-s + (0.327 − 0.945i)4-s + (0.959 − 0.281i)5-s + (0.165 + 0.986i)6-s + (0.999 − 0.0237i)7-s + (−0.281 − 0.959i)8-s + (−0.618 − 0.786i)9-s + (0.618 − 0.786i)10-s + (−0.690 − 0.723i)11-s + (0.707 + 0.707i)12-s + (0.800 − 0.599i)14-s + (−0.165 + 0.986i)15-s + (−0.786 − 0.618i)16-s + (0.814 + 0.580i)17-s + (−0.959 − 0.281i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.542 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.542 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.436123187 - 1.327627546i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.436123187 - 1.327627546i\) |
\(L(1)\) |
\(\approx\) |
\(1.749955086 - 0.5074011490i\) |
\(L(1)\) |
\(\approx\) |
\(1.749955086 - 0.5074011490i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
| 89 | \( 1 \) |
good | 2 | \( 1 + (0.814 - 0.580i)T \) |
| 3 | \( 1 + (-0.436 + 0.899i)T \) |
| 5 | \( 1 + (0.959 - 0.281i)T \) |
| 7 | \( 1 + (0.999 - 0.0237i)T \) |
| 11 | \( 1 + (-0.690 - 0.723i)T \) |
| 17 | \( 1 + (0.814 + 0.580i)T \) |
| 19 | \( 1 + (0.992 + 0.118i)T \) |
| 23 | \( 1 + (0.393 + 0.919i)T \) |
| 29 | \( 1 + (-0.520 - 0.853i)T \) |
| 31 | \( 1 + (-0.800 + 0.599i)T \) |
| 37 | \( 1 + (-0.965 + 0.258i)T \) |
| 41 | \( 1 + (0.560 - 0.828i)T \) |
| 43 | \( 1 + (0.520 - 0.853i)T \) |
| 47 | \( 1 + (-0.654 - 0.755i)T \) |
| 53 | \( 1 + (0.755 + 0.654i)T \) |
| 59 | \( 1 + (-0.899 + 0.436i)T \) |
| 61 | \( 1 + (0.739 - 0.672i)T \) |
| 67 | \( 1 + (-0.945 + 0.327i)T \) |
| 71 | \( 1 + (-0.235 - 0.971i)T \) |
| 73 | \( 1 + (0.989 + 0.142i)T \) |
| 79 | \( 1 + (-0.989 - 0.142i)T \) |
| 83 | \( 1 + (0.936 + 0.349i)T \) |
| 97 | \( 1 + (0.690 - 0.723i)T \) |
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
−21.47811006647154804404612026345, −20.78472579078601968706753801479, −20.20813174698552332206059516942, −18.68977878791829233428135542169, −18.00857745184465127806426176125, −17.71401846122257200769859271345, −16.76821821412274737263472186109, −16.1240431788840709751494233812, −14.74551480361717479074843048246, −14.44205238989986285431652627005, −13.61140426706129908949078726924, −12.903517974110830423804347888166, −12.24947948489750390481166019806, −11.32707381348520448234031859239, −10.668182074352218301250009580787, −9.398883811795178474179431790949, −8.24748543449478964276748491090, −7.43505417012054995245205917380, −6.99370808356481238827127251081, −5.85833674218162802542465059141, −5.29092064061368048200822916926, −4.70197964670639824008003727780, −3.05427745319780539581126536271, −2.28449468547575727739131879572, −1.38311022151609833671519655901,
0.97550993094864773449085357313, 1.95227573563816598773577682010, 3.12553722017391096522353733463, 3.90733929915604425044142528201, 5.12129337792010114511860727811, 5.39074675916601984766142571676, 6.008047801952904466951332486298, 7.40351073595527027167983583554, 8.70008686932320107133005345528, 9.504817547612865920440065071444, 10.37192035503994331870121523744, 10.82526825040077339259755917976, 11.69341493251124055446802826774, 12.40667938363663622718126351066, 13.54114748654079105271201596951, 14.023010096672223868314926772887, 14.83493534136830320954884889703, 15.592540082916582556069663173362, 16.445959301482659442378564129517, 17.25772353495749770658468051373, 18.04823395146067173970608554930, 18.85130920385660604346025129542, 20.03641610528614644932713653098, 20.87029273499862719842509528956, 21.13895912055851984543814440420