| L(s) = 1 | + 0.356i·5-s + 7-s + 5.96i·11-s − 2.87i·13-s − 3.16·17-s + 0.127i·19-s − 2.80·23-s + 4.87·25-s − 2.44i·29-s + 7·31-s + 0.356i·35-s + 1.87i·37-s + 6.99·41-s + 1.12i·43-s − 7.34·47-s + ⋯ |
| L(s) = 1 | + 0.159i·5-s + 0.377·7-s + 1.79i·11-s − 0.796i·13-s − 0.766·17-s + 0.0291i·19-s − 0.585·23-s + 0.974·25-s − 0.454i·29-s + 1.25·31-s + 0.0602i·35-s + 0.307i·37-s + 1.09·41-s + 0.171i·43-s − 1.07·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.126 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.591184479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.591184479\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 0.356iT - 5T^{2} \) |
| 11 | \( 1 - 5.96iT - 11T^{2} \) |
| 13 | \( 1 + 2.87iT - 13T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 19 | \( 1 - 0.127iT - 19T^{2} \) |
| 23 | \( 1 + 2.80T + 23T^{2} \) |
| 29 | \( 1 + 2.44iT - 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 - 1.87iT - 37T^{2} \) |
| 41 | \( 1 - 6.99T + 41T^{2} \) |
| 43 | \( 1 - 1.12iT - 43T^{2} \) |
| 47 | \( 1 + 7.34T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 6.63iT - 59T^{2} \) |
| 61 | \( 1 - 13.7iT - 61T^{2} \) |
| 67 | \( 1 + 8.61iT - 67T^{2} \) |
| 71 | \( 1 - 5.96T + 71T^{2} \) |
| 73 | \( 1 - 0.872T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 - 6.63iT - 83T^{2} \) |
| 89 | \( 1 - 6.99T + 89T^{2} \) |
| 97 | \( 1 + 9.74T + 97T^{2} \) |
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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
−8.093211199314389278333853128576, −7.63314563329950704052110570434, −6.84900186573982486967947196842, −6.27551135068909372460088274197, −5.27077150608199425768713653567, −4.61117609079899404907317415697, −4.07229270836031200702358608344, −2.82424016999826177957495345324, −2.20619947455441116960769008433, −1.12485555268340234170082299514,
0.43441712241863975160851200579, 1.50629415391339233863319085576, 2.58596582219321154984449818660, 3.40061881681961208076739779359, 4.28056682474232988966333478028, 4.95115991947633861246806724770, 5.83053011585883655796712076730, 6.42652262407337434144881552150, 7.08419048170983042816042645334, 8.174377951074789952995580403480
