L(s) = 1 | + (−1.75 + 2.21i)2-s + (−3.44 − 5.96i)3-s + (−1.81 − 7.79i)4-s + (−4.17 − 2.41i)5-s + (19.2 + 2.85i)6-s + (−5.03 − 17.8i)7-s + (20.4 + 9.67i)8-s + (−10.1 + 17.6i)9-s + (12.6 − 5.01i)10-s + (−36.6 + 21.1i)11-s + (−40.1 + 37.6i)12-s − 3.39i·13-s + (48.3 + 20.1i)14-s + 33.1i·15-s + (−57.4 + 28.2i)16-s + (101. − 58.6i)17-s + ⋯ |
L(s) = 1 | + (−0.621 + 0.783i)2-s + (−0.662 − 1.14i)3-s + (−0.227 − 0.973i)4-s + (−0.373 − 0.215i)5-s + (1.31 + 0.194i)6-s + (−0.272 − 0.962i)7-s + (0.903 + 0.427i)8-s + (−0.377 + 0.653i)9-s + (0.401 − 0.158i)10-s + (−1.00 + 0.579i)11-s + (−0.966 + 0.905i)12-s − 0.0724i·13-s + (0.922 + 0.385i)14-s + 0.571i·15-s + (−0.896 + 0.442i)16-s + (1.45 − 0.837i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.145 + 0.989i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.145 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.344141 - 0.398365i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.344141 - 0.398365i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.75 - 2.21i)T \) |
| 7 | \( 1 + (5.03 + 17.8i)T \) |
good | 3 | \( 1 + (3.44 + 5.96i)T + (-13.5 + 23.3i)T^{2} \) |
| 5 | \( 1 + (4.17 + 2.41i)T + (62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (36.6 - 21.1i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 3.39iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-101. + 58.6i)T + (2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-45.5 + 78.9i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-147. - 85.3i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 131.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-5.70 - 9.87i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-59.2 + 102. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 109. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 82.5iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (36.7 - 63.6i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-87.2 - 151. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (166. + 288. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-472. - 272. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-516. + 298. i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 384. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-187. + 108. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-868. - 501. i)T + (2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 459.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (771. + 445. i)T + (3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 282. iT - 9.12e5T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.66090880278139353717377629651, −15.51896813647346939830191806079, −13.84958797959523213306673928025, −12.79329697569113844914251686732, −11.23405589865359834891578816763, −9.744848240467669435481798420280, −7.65721701120609619698717785046, −7.10339578309585769637737070374, −5.31463744927972684246645694688, −0.69573783523621706237239359132,
3.39341462708832315047572350899, 5.42060405145251106521295249203, 8.058036568567065420445067343246, 9.597085680958902652942478299635, 10.61378438341777872879413515167, 11.62689784517232674193305890464, 12.86388487013652917903452975426, 15.00993336321250934082767426176, 16.18192043747860157805657567695, 16.88713470804962148212964341840