L(s) = 1 | − 21.0·5-s + 25.8i·7-s + 46.9i·11-s + 55.4i·13-s + 64.8i·17-s + 51.0·19-s − 60.4·23-s + 316.·25-s − 90.5·29-s − 8.35i·31-s − 542. i·35-s + 228. i·37-s + 239. i·41-s + 192.·43-s − 38.1·47-s + ⋯ |
L(s) = 1 | − 1.88·5-s + 1.39i·7-s + 1.28i·11-s + 1.18i·13-s + 0.924i·17-s + 0.616·19-s − 0.548·23-s + 2.53·25-s − 0.579·29-s − 0.0484i·31-s − 2.62i·35-s + 1.01i·37-s + 0.913i·41-s + 0.684·43-s − 0.118·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.829 + 0.558i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.829 + 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.7612643001\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7612643001\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 21.0T + 125T^{2} \) |
| 7 | \( 1 - 25.8iT - 343T^{2} \) |
| 11 | \( 1 - 46.9iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 55.4iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 64.8iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 51.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 60.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 90.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 8.35iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 228. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 239. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 192.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 38.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 6.69T + 1.48e5T^{2} \) |
| 59 | \( 1 - 5.40iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 847. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 992.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 494.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 395.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 42.1iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 328. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.13e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 991.T + 9.12e5T^{2} \) |
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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
−10.28183946665015918359576192020, −9.225115278610879241944291306447, −8.584543055471530604335238061480, −7.73124738796394369576486645969, −7.05481368561931252473531659636, −5.97491626605029585245017621309, −4.69050656649796753794640472016, −4.12014062310124817517961086413, −2.94339968577417363972331300211, −1.67212888633887536069600831579,
0.30739054170250700587164994447, 0.75407127658672776293419635582, 3.15023171176645699565524372311, 3.65811981647020855935418947199, 4.55738476910285831395709856306, 5.69369409190069118209442628701, 7.07634197539708309943091584245, 7.61422956264418885565150353469, 8.139824363925271207069153227955, 9.143950525649357253798544790682