| L(s) = 1 | + (0.597 − 1.24i)2-s + (8.51 + 0.638i)3-s + (8.79 + 11.0i)4-s + (−9.70 + 31.4i)5-s + (5.88 − 10.1i)6-s + (−17.8 + 10.2i)7-s + (40.4 − 9.23i)8-s + (−7.93 − 1.19i)9-s + (33.2 + 30.8i)10-s + (73.8 − 92.6i)11-s + (67.8 + 99.5i)12-s + (106. − 98.5i)13-s + (2.11 + 28.2i)14-s + (−102. + 261. i)15-s + (−37.4 + 164. i)16-s + (188. − 58.1i)17-s + ⋯ |
| L(s) = 1 | + (0.149 − 0.310i)2-s + (0.946 + 0.0709i)3-s + (0.549 + 0.689i)4-s + (−0.388 + 1.25i)5-s + (0.163 − 0.283i)6-s + (−0.363 + 0.209i)7-s + (0.631 − 0.144i)8-s + (−0.0979 − 0.0147i)9-s + (0.332 + 0.308i)10-s + (0.610 − 0.765i)11-s + (0.471 + 0.691i)12-s + (0.628 − 0.583i)13-s + (0.0108 + 0.144i)14-s + (−0.456 + 1.16i)15-s + (−0.146 + 0.641i)16-s + (0.652 − 0.201i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 - 0.557i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.830 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.07896 + 0.633074i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.07896 + 0.633074i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.39e3 + 1.20e3i)T \) |
| good | 2 | \( 1 + (-0.597 + 1.24i)T + (-9.97 - 12.5i)T^{2} \) |
| 3 | \( 1 + (-8.51 - 0.638i)T + (80.0 + 12.0i)T^{2} \) |
| 5 | \( 1 + (9.70 - 31.4i)T + (-516. - 352. i)T^{2} \) |
| 7 | \( 1 + (17.8 - 10.2i)T + (1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-73.8 + 92.6i)T + (-3.25e3 - 1.42e4i)T^{2} \) |
| 13 | \( 1 + (-106. + 98.5i)T + (2.13e3 - 2.84e4i)T^{2} \) |
| 17 | \( 1 + (-188. + 58.1i)T + (6.90e4 - 4.70e4i)T^{2} \) |
| 19 | \( 1 + (94.2 + 625. i)T + (-1.24e5 + 3.84e4i)T^{2} \) |
| 23 | \( 1 + (-4.42 - 11.2i)T + (-2.05e5 + 1.90e5i)T^{2} \) |
| 29 | \( 1 + (-226. + 16.9i)T + (6.99e5 - 1.05e5i)T^{2} \) |
| 31 | \( 1 + (1.07e3 - 731. i)T + (3.37e5 - 8.59e5i)T^{2} \) |
| 37 | \( 1 + (-802. - 463. i)T + (9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + (-1.12e3 - 542. i)T + (1.76e6 + 2.20e6i)T^{2} \) |
| 47 | \( 1 + (-1.27e3 - 1.60e3i)T + (-1.08e6 + 4.75e6i)T^{2} \) |
| 53 | \( 1 + (1.54e3 + 1.43e3i)T + (5.89e5 + 7.86e6i)T^{2} \) |
| 59 | \( 1 + (-654. + 2.86e3i)T + (-1.09e7 - 5.25e6i)T^{2} \) |
| 61 | \( 1 + (-1.61e3 + 2.37e3i)T + (-5.05e6 - 1.28e7i)T^{2} \) |
| 67 | \( 1 + (3.62e3 - 546. i)T + (1.92e7 - 5.93e6i)T^{2} \) |
| 71 | \( 1 + (649. + 254. i)T + (1.86e7 + 1.72e7i)T^{2} \) |
| 73 | \( 1 + (5.06e3 + 5.45e3i)T + (-2.12e6 + 2.83e7i)T^{2} \) |
| 79 | \( 1 + (4.52e3 + 7.83e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (896. - 1.19e4i)T + (-4.69e7 - 7.07e6i)T^{2} \) |
| 89 | \( 1 + (-1.23e4 - 927. i)T + (6.20e7 + 9.35e6i)T^{2} \) |
| 97 | \( 1 + (7.75e3 - 9.72e3i)T + (-1.96e7 - 8.63e7i)T^{2} \) |
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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
−15.19860828216019731816573233966, −14.19282031689943024938431181195, −13.08610344443840531360806685692, −11.55838025838246621421189005421, −10.79321414897398833342683189673, −8.987015351837616898874898914369, −7.74129549156668127310886279877, −6.49692496006874490013842096467, −3.47025303297834630753365968951, −2.83806383490033856514843652539,
1.59208746962976586097830619982, 4.05610772645766538995363358416, 5.85721600368129895072131811048, 7.56380076010047053306795218281, 8.784407270191070001854436656520, 9.952270565086768466769125553990, 11.69006334437710759395875656598, 12.90102688584369709913430700901, 14.20670443481468732687163890829, 14.91912111427393419368495081753
