L(s) = 1 | + 15.2i·3-s − 43.0i·5-s + 22.6·7-s + 10.9·9-s + 258. i·11-s − 990. i·13-s + 656.·15-s − 218·17-s − 1.99e3i·19-s + 344. i·21-s + 3.41e3·23-s + 1.26e3·25-s + 3.86e3i·27-s + 4.86e3i·29-s + 8.68e3·31-s + ⋯ |
L(s) = 1 | + 0.977i·3-s − 0.770i·5-s + 0.174·7-s + 0.0452·9-s + 0.645i·11-s − 1.62i·13-s + 0.753·15-s − 0.182·17-s − 1.26i·19-s + 0.170i·21-s + 1.34·23-s + 0.406·25-s + 1.02i·27-s + 1.07i·29-s + 1.62·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.003458295\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.003458295\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 - 15.2iT - 243T^{2} \) |
| 5 | \( 1 + 43.0iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 22.6T + 1.68e4T^{2} \) |
| 11 | \( 1 - 258. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 990. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 218T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.99e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 3.41e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.86e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 8.68e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.23e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 8.41e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 685. iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.81e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.65e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.37e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 2.73e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 6.67e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 2.16e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.30e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.47e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.06e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 9.38e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.17e4T + 8.58e9T^{2} \) |
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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
−12.63306500601536291018574124275, −11.16253304555176162793338714902, −10.28967070238631239695101632358, −9.330605120610664915857295549104, −8.378951727488655374745141279349, −7.00226285533462254063451757227, −5.17916943069503227562521706024, −4.63087958265224219231533912176, −3.02257780166268308549100242413, −0.895199642410913806210658281989,
1.19495273826889767171684774228, 2.57282118404588564069910509778, 4.24852247619541758845156518829, 6.13149051378128802698711067550, 6.89975930597356973392711698791, 7.900178117428152812110260591625, 9.165338446888896948635722972706, 10.50285769267118439890769358039, 11.52412622534506982988867633609, 12.34712771300040917737510041229