L(s) = 1 | + (4.08 + 8.01i)3-s + 47.0i·5-s + 13.4·7-s + (−47.6 + 65.5i)9-s − 135. i·11-s − 80.8·13-s + (−377. + 192. i)15-s + 430. i·17-s − 43.5·19-s + (54.9 + 107. i)21-s + 873. i·23-s − 1.59e3·25-s + (−719. − 114. i)27-s − 707. i·29-s + 677.·31-s + ⋯ |
L(s) = 1 | + (0.453 + 0.891i)3-s + 1.88i·5-s + 0.274·7-s + (−0.587 + 0.808i)9-s − 1.11i·11-s − 0.478·13-s + (−1.67 + 0.854i)15-s + 1.49i·17-s − 0.120·19-s + (0.124 + 0.244i)21-s + 1.65i·23-s − 2.54·25-s + (−0.987 − 0.156i)27-s − 0.840i·29-s + 0.705·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.537940490\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.537940490\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.08 - 8.01i)T \) |
good | 5 | \( 1 - 47.0iT - 625T^{2} \) |
| 7 | \( 1 - 13.4T + 2.40e3T^{2} \) |
| 11 | \( 1 + 135. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 80.8T + 2.85e4T^{2} \) |
| 17 | \( 1 - 430. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 43.5T + 1.30e5T^{2} \) |
| 23 | \( 1 - 873. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 707. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 677.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.63e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.22e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.43e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.06e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.46e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 470. iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 39.9T + 1.38e7T^{2} \) |
| 67 | \( 1 - 4.52e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 9.85e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 3.27e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 6.93e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.27e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 9.78e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 9.32e3T + 8.85e7T^{2} \) |
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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
−11.15375035299656877029616514985, −10.29791969667859373217323045488, −9.723622674805926597969806814534, −8.372046268028204994163306373642, −7.66549606205066266625989986340, −6.42115488499179082405012822944, −5.55832139721510692709064046271, −3.92726797305344805878189639245, −3.25354840009936477173801134772, −2.18216816317090146436935925117,
0.40158405848014785282479756981, 1.41963700355927580300746603216, 2.58340127032561573640364065575, 4.48914950197873528244113195884, 5.03425095800699122626469277667, 6.47051087662206637987488721581, 7.58341473896419752409732953913, 8.291660805348687748614919420580, 9.171049290774593672266010918611, 9.780049953932637177808533559591