L(s) = 1 | − 2-s + (−0.5 + 0.866i)3-s + 4-s + (−0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s + (0.5 + 0.866i)7-s − 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (−0.5 + 0.866i)12-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)15-s + 16-s + (0.5 + 0.866i)17-s + (0.5 + 0.866i)18-s + ⋯ |
L(s) = 1 | − 2-s + (−0.5 + 0.866i)3-s + 4-s + (−0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s + (0.5 + 0.866i)7-s − 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (−0.5 + 0.866i)12-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)15-s + 16-s + (0.5 + 0.866i)17-s + (0.5 + 0.866i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.736 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.736 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.2616066818 + 0.6722047101i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.2616066818 + 0.6722047101i\) |
\(L(1)\) |
\(\approx\) |
\(0.4257981876 + 0.3905783447i\) |
\(L(1)\) |
\(\approx\) |
\(0.4257981876 + 0.3905783447i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 43 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.5 + 0.866i)T \) |
| 17 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (-0.5 + 0.866i)T \) |
| 41 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (-0.5 + 0.866i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + (0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 + (0.5 + 0.866i)T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 + (0.5 - 0.866i)T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + T \) |
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
−23.375100787612000049212917418900, −22.53854546750660089973316350118, −20.7671232674662437250379680083, −20.49672089798646656076659669693, −19.618059593208894769023887500545, −18.68140571145159113332149165695, −17.96472656730433086616311730745, −17.13916008881439817462951238940, −16.49412447598268557852949251474, −15.784206434684267711559568651786, −14.39477703336415988551929181392, −13.33200530662796375070057168754, −12.31383264399294263619790572412, −11.67893200118759170337015093618, −10.79175514533776973451313363276, −9.87753595444631835175169115537, −8.49290102441012553765058891268, −7.8819976136288738186359867214, −7.289616729288237635392646104319, −6.04308965598426468220746230086, −5.08805193780819795480191090219, −3.5942877266344139762105901297, −2.01680032057906862805830955022, −0.90620473456205608834243253926, −0.37219326722839659227260692944,
1.43825863504808189283627950684, 2.84658740411653765640893107602, 3.78431548707882221488747441356, 5.25674127137172546170285902344, 6.244527462251536049054034541978, 7.13058614421734165520151541124, 8.38240311352401932286037063187, 9.06794152630538219389900384576, 10.12972455347962294727610999487, 10.89253991236040474521010514376, 11.63256797145207887326634571276, 12.15257380667888996660954792026, 14.1639259677572629761234924946, 15.050372581081766116089932654381, 15.637034332868146407324419863567, 16.33753387247630803632489353943, 17.45706464164734552093713405061, 18.09640654676364861929859963755, 18.882433104415866964003879889478, 19.74640159216074315152878870645, 20.7745604116122750203854398712, 21.73988253760448933232236443684, 22.0073332806596566132117825855, 23.62849346660906155344360403846, 23.85692144465160884422781471438