L(s) = 1 | + (−0.618 + 1.61i)3-s − 1.23·5-s + 3.23i·7-s + (−2.23 − 2.00i)9-s + 0.763i·11-s + 4.47i·13-s + (0.763 − 2.00i)15-s − 6.47i·17-s − 5.23·19-s + (−5.23 − 2.00i)21-s + 6.47·23-s − 3.47·25-s + (4.61 − 2.38i)27-s − 9.23·29-s + 0.763i·31-s + ⋯ |
L(s) = 1 | + (−0.356 + 0.934i)3-s − 0.552·5-s + 1.22i·7-s + (−0.745 − 0.666i)9-s + 0.230i·11-s + 1.24i·13-s + (0.197 − 0.516i)15-s − 1.56i·17-s − 1.20·19-s + (−1.14 − 0.436i)21-s + 1.34·23-s − 0.694·25-s + (0.888 − 0.458i)27-s − 1.71·29-s + 0.137i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 + 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0921803 - 0.431916i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0921803 - 0.431916i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.618 - 1.61i)T \) |
good | 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 - 3.23iT - 7T^{2} \) |
| 11 | \( 1 - 0.763iT - 11T^{2} \) |
| 13 | \( 1 - 4.47iT - 13T^{2} \) |
| 17 | \( 1 + 6.47iT - 17T^{2} \) |
| 19 | \( 1 + 5.23T + 19T^{2} \) |
| 23 | \( 1 - 6.47T + 23T^{2} \) |
| 29 | \( 1 + 9.23T + 29T^{2} \) |
| 31 | \( 1 - 0.763iT - 31T^{2} \) |
| 37 | \( 1 - 0.472iT - 37T^{2} \) |
| 41 | \( 1 + 2.47iT - 41T^{2} \) |
| 43 | \( 1 + 2.76T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 1.23T + 53T^{2} \) |
| 59 | \( 1 - 3.23iT - 59T^{2} \) |
| 61 | \( 1 + 8.47iT - 61T^{2} \) |
| 67 | \( 1 - 3.70T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 13.7iT - 79T^{2} \) |
| 83 | \( 1 + 7.23iT - 83T^{2} \) |
| 89 | \( 1 - 4iT - 89T^{2} \) |
| 97 | \( 1 + 8.47T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.05930996089870945074026964490, −9.738315316494419976777170980624, −9.178469164861366366257637682500, −8.585964302788056660269182102667, −7.27426726825701418649957522303, −6.33311189130423687249434381240, −5.28054154575686224755064032135, −4.58007398019946576177703920396, −3.50423865323254091183318992841, −2.27226121625915783222418550247,
0.22898443236149848169555542180, 1.61921106160247963726547859665, 3.26251381525882342521707204672, 4.24888606900431887658516334376, 5.53713211455925263401734154656, 6.40942945489643600095796549278, 7.34601987751179902385258766770, 7.937518760407623747524673603422, 8.652782704242151537695459857454, 10.14172047491469205434816040678