L(s) = 1 | + (0.137 − 0.990i)2-s + (0.583 − 0.812i)3-s + (−0.962 − 0.272i)4-s + (−0.480 + 0.876i)5-s + (−0.724 − 0.689i)6-s + (0.601 + 0.799i)7-s + (−0.401 + 0.915i)8-s + (−0.319 − 0.947i)9-s + (0.802 + 0.596i)10-s + (−0.917 − 0.396i)11-s + (−0.782 + 0.622i)12-s + (0.775 − 0.631i)13-s + (0.874 − 0.485i)14-s + (0.431 + 0.901i)15-s + (0.851 + 0.523i)16-s + (−0.381 − 0.924i)17-s + ⋯ |
L(s) = 1 | + (0.137 − 0.990i)2-s + (0.583 − 0.812i)3-s + (−0.962 − 0.272i)4-s + (−0.480 + 0.876i)5-s + (−0.724 − 0.689i)6-s + (0.601 + 0.799i)7-s + (−0.401 + 0.915i)8-s + (−0.319 − 0.947i)9-s + (0.802 + 0.596i)10-s + (−0.917 − 0.396i)11-s + (−0.782 + 0.622i)12-s + (0.775 − 0.631i)13-s + (0.874 − 0.485i)14-s + (0.431 + 0.901i)15-s + (0.851 + 0.523i)16-s + (−0.381 − 0.924i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.744 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.744 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5059214034 - 1.320718133i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5059214034 - 1.320718133i\) |
\(L(1)\) |
\(\approx\) |
\(0.8699840860 - 0.7684208603i\) |
\(L(1)\) |
\(\approx\) |
\(0.8699840860 - 0.7684208603i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 571 | \( 1 \) |
good | 2 | \( 1 + (0.137 - 0.990i)T \) |
| 3 | \( 1 + (0.583 - 0.812i)T \) |
| 5 | \( 1 + (-0.480 + 0.876i)T \) |
| 7 | \( 1 + (0.601 + 0.799i)T \) |
| 11 | \( 1 + (-0.917 - 0.396i)T \) |
| 13 | \( 1 + (0.775 - 0.631i)T \) |
| 17 | \( 1 + (-0.381 - 0.924i)T \) |
| 19 | \( 1 + (0.952 + 0.303i)T \) |
| 23 | \( 1 + (0.746 - 0.665i)T \) |
| 29 | \( 1 + (0.451 - 0.892i)T \) |
| 31 | \( 1 + (-0.879 + 0.475i)T \) |
| 37 | \( 1 + (0.840 - 0.542i)T \) |
| 41 | \( 1 + (0.904 - 0.426i)T \) |
| 43 | \( 1 + (-0.999 - 0.0110i)T \) |
| 47 | \( 1 + (0.0275 - 0.999i)T \) |
| 53 | \( 1 + (0.471 - 0.882i)T \) |
| 59 | \( 1 + (0.945 - 0.324i)T \) |
| 61 | \( 1 + (-0.644 - 0.764i)T \) |
| 67 | \( 1 + (0.528 + 0.849i)T \) |
| 71 | \( 1 + (-0.978 - 0.207i)T \) |
| 73 | \( 1 + (-0.889 - 0.456i)T \) |
| 79 | \( 1 + (-0.441 + 0.897i)T \) |
| 83 | \( 1 + (0.959 - 0.282i)T \) |
| 89 | \( 1 + (-0.234 + 0.972i)T \) |
| 97 | \( 1 + (0.938 - 0.345i)T \) |
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
−23.671496428886538322784935880736, −23.0671058039387079922341615500, −21.80329119445496096118782199806, −21.119668491812219332800550120015, −20.365099459789835593115439582839, −19.58859768871348228050853680789, −18.386930181481189945419282475682, −17.40895476589464050424033118638, −16.56246555633565743777491415122, −16.01088705612033166772457186447, −15.23952650708123379458528575903, −14.48462524392123920869987804421, −13.41716525788472798589022846691, −13.057855187845221610098709711009, −11.512920287647796098475891733822, −10.53381195319425904624716431569, −9.414892042629777943176002275848, −8.70363712230759315946381169625, −7.89265143430201619016457685368, −7.27225890730275080540206705207, −5.6781938357791534902059460980, −4.75419136645226857188428290212, −4.25565908863516521764508868765, −3.28081333990902112956804991904, −1.35182550415572144289807993526,
0.7279664670832002416565303613, 2.20720635752100829734828014705, 2.83172774045046792176359711387, 3.619322395671140428152517946365, 5.10943174990763403255500717718, 6.06063923182754997232170064251, 7.42943397635593499755888056946, 8.22761799762520497630803534918, 8.94054366984959077900485092166, 10.15770010944821783316943606596, 11.21563291697963836339626856129, 11.66670562809548314280171195794, 12.70171673966263127313025899876, 13.48423040980318116500637674212, 14.29616679920718293154622485806, 15.02196613387591910954126033254, 15.92131564324870516339544819035, 17.83148829409006385232507677294, 18.26540529110084529816429580514, 18.64761127658883879994872156818, 19.55209524187357509135142394314, 20.473808401123099387592283119503, 21.032810563765586279176162650678, 22.06121126712284939868352989259, 22.98478674240768642534214400352