L(s) = 1 | + (0.618 + 1.61i)3-s − i·5-s − 2i·7-s + (−2.23 + 2.00i)9-s − 2.47·11-s + 1.23·13-s + (1.61 − 0.618i)15-s + 0.763i·17-s + 5.23i·19-s + (3.23 − 1.23i)21-s − 0.472·23-s − 25-s + (−4.61 − 2.38i)27-s + 8.47i·29-s + 4.76i·31-s + ⋯ |
L(s) = 1 | + (0.356 + 0.934i)3-s − 0.447i·5-s − 0.755i·7-s + (−0.745 + 0.666i)9-s − 0.745·11-s + 0.342·13-s + (0.417 − 0.159i)15-s + 0.185i·17-s + 1.20i·19-s + (0.706 − 0.269i)21-s − 0.0984·23-s − 0.200·25-s + (−0.888 − 0.458i)27-s + 1.57i·29-s + 0.855i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.414245059\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.414245059\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.618 - 1.61i)T \) |
| 5 | \( 1 + iT \) |
good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 2.47T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 - 0.763iT - 17T^{2} \) |
| 19 | \( 1 - 5.23iT - 19T^{2} \) |
| 23 | \( 1 + 0.472T + 23T^{2} \) |
| 29 | \( 1 - 8.47iT - 29T^{2} \) |
| 31 | \( 1 - 4.76iT - 31T^{2} \) |
| 37 | \( 1 - 7.70T + 37T^{2} \) |
| 41 | \( 1 - 1.52iT - 41T^{2} \) |
| 43 | \( 1 - 9.70iT - 43T^{2} \) |
| 47 | \( 1 + 4.47T + 47T^{2} \) |
| 53 | \( 1 - 4.47iT - 53T^{2} \) |
| 59 | \( 1 + 6.47T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 11.2iT - 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 0.472T + 73T^{2} \) |
| 79 | \( 1 - 8.18iT - 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 1.52iT - 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.556359215218261213709673152953, −8.590450216732712507591849571892, −8.098392325599929305513472703853, −7.27827030610481072296816854187, −6.08615721285297465158960794782, −5.26846903351067718903430469572, −4.47800868929200539881350424434, −3.71014236957472017396051927018, −2.82269707886158081340630212193, −1.36848107268474997155698320135,
0.49516445494296147655176957050, 2.20574796415090258682462588150, 2.62971272531666290101872995691, 3.76733838760474193915933214850, 5.07034953409739136830547364617, 5.97079536289678160479837008754, 6.56065395630827630856596101024, 7.52969399254848029228637578202, 8.031159511394821690628683966677, 8.906781570872534912173976534568