L(s) = 1 | + (−0.304 − 1.38i)2-s + (−1.81 + 0.842i)4-s − 0.556i·5-s + (−1.25 − 2.33i)7-s + (1.71 + 2.24i)8-s + (−0.769 + 0.169i)10-s − 0.384i·11-s − 4.88i·13-s + (−2.83 + 2.43i)14-s + (2.58 − 3.05i)16-s + 5.55i·17-s − 5.79·19-s + (0.469 + 1.01i)20-s + (−0.530 + 0.117i)22-s − 2.42i·23-s + ⋯ |
L(s) = 1 | + (−0.215 − 0.976i)2-s + (−0.907 + 0.421i)4-s − 0.249i·5-s + (−0.473 − 0.880i)7-s + (0.606 + 0.794i)8-s + (−0.243 + 0.0537i)10-s − 0.115i·11-s − 1.35i·13-s + (−0.758 + 0.652i)14-s + (0.645 − 0.763i)16-s + 1.34i·17-s − 1.32·19-s + (0.104 + 0.225i)20-s + (−0.113 + 0.0249i)22-s − 0.505i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.800 - 0.599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.152990 + 0.459268i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.152990 + 0.459268i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.304 + 1.38i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.25 + 2.33i)T \) |
good | 5 | \( 1 + 0.556iT - 5T^{2} \) |
| 11 | \( 1 + 0.384iT - 11T^{2} \) |
| 13 | \( 1 + 4.88iT - 13T^{2} \) |
| 17 | \( 1 - 5.55iT - 17T^{2} \) |
| 19 | \( 1 + 5.79T + 19T^{2} \) |
| 23 | \( 1 + 2.42iT - 23T^{2} \) |
| 29 | \( 1 + 6.72T + 29T^{2} \) |
| 31 | \( 1 + 8.00T + 31T^{2} \) |
| 37 | \( 1 + 4.16T + 37T^{2} \) |
| 41 | \( 1 - 7.47iT - 41T^{2} \) |
| 43 | \( 1 + 7.80iT - 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 3.43T + 53T^{2} \) |
| 59 | \( 1 - 1.47T + 59T^{2} \) |
| 61 | \( 1 - 3.06iT - 61T^{2} \) |
| 67 | \( 1 + 1.91iT - 67T^{2} \) |
| 71 | \( 1 + 3.10iT - 71T^{2} \) |
| 73 | \( 1 + 4.89iT - 73T^{2} \) |
| 79 | \( 1 + 6.35iT - 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 7.07iT - 89T^{2} \) |
| 97 | \( 1 + 9.28iT - 97T^{2} \) |
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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
−10.19954666386326882120138276957, −9.022777092429336500436007083207, −8.331785091386251069706646178575, −7.47685009885414315914509625860, −6.23651469316019687295029128852, −5.08459349573095168176395007160, −3.97630873581715661347625116385, −3.25201844999479314599431537320, −1.76840362722994967675903759069, −0.25907876972168930767456108406,
2.04511014659137749388552001974, 3.61176625531413791336878913031, 4.79584177901727162429472014154, 5.65900964040623086473750902764, 6.69678232601449730199832073125, 7.12197638046900522072532307060, 8.351915696075550670281073165634, 9.247709473717351804974429599095, 9.477128512941213231352625985686, 10.76206916133635871402793852563