L(s) = 1 | + (−4.64 − 8.04i)3-s + (17.1 + 9.92i)5-s + (−18.3 + 2.13i)7-s + (−29.6 + 51.4i)9-s + (10.2 − 5.91i)11-s − 19.9i·13-s − 184. i·15-s + (1.57 − 0.908i)17-s + (3.66 − 6.34i)19-s + (102. + 138. i)21-s + (92.2 + 53.2i)23-s + (134. + 233. i)25-s + 300.·27-s + 191.·29-s + (62.5 + 108. i)31-s + ⋯ |
L(s) = 1 | + (−0.894 − 1.54i)3-s + (1.53 + 0.887i)5-s + (−0.993 + 0.115i)7-s + (−1.09 + 1.90i)9-s + (0.280 − 0.162i)11-s − 0.426i·13-s − 3.17i·15-s + (0.0224 − 0.0129i)17-s + (0.0442 − 0.0765i)19-s + (1.06 + 1.43i)21-s + (0.835 + 0.482i)23-s + (1.07 + 1.86i)25-s + 2.14·27-s + 1.22·29-s + (0.362 + 0.627i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.490 + 0.871i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.629098494\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.629098494\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (18.3 - 2.13i)T \) |
good | 3 | \( 1 + (4.64 + 8.04i)T + (-13.5 + 23.3i)T^{2} \) |
| 5 | \( 1 + (-17.1 - 9.92i)T + (62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-10.2 + 5.91i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 19.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-1.57 + 0.908i)T + (2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-3.66 + 6.34i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-92.2 - 53.2i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 191.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-62.5 - 108. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-158. + 274. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 321. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 74.3iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-77.2 + 133. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (159. + 276. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-30.5 - 52.9i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-267. - 154. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-514. + 297. i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 48.6iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (667. - 385. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-831. - 480. i)T + (2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 1.06e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (567. + 327. i)T + (3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 704. iT - 9.12e5T^{2} \) |
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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
−10.59833950075933939720038500687, −9.837240060376346754789826873626, −8.711386460318045274218107138211, −7.21377126854606517807982300458, −6.75307719884275887779280495186, −5.97909883854942806910482581879, −5.40995046895051254049890331506, −3.02225681362660782168733489126, −2.09200628111468260606438664574, −0.802018566437416210893770641774,
0.929313392384516992040831821713, 2.87679558991654803057034636555, 4.35991352752197713308002926062, 4.99178441908734743634873291268, 6.06857307944276772853227938914, 6.48424483978681993969082033807, 8.648277147957012557016749558346, 9.478290779603464221184626927809, 9.812028034536835551211124213489, 10.50555258498914494072615662174