| L(s) = 1 | + (−558. − 461. i)2-s + (−9.66e3 + 9.66e3i)3-s + (9.89e4 + 5.14e5i)4-s + (−2.13e6 − 3.81e6i)5-s + (9.85e6 − 9.38e5i)6-s + (4.16e7 + 4.16e7i)7-s + (1.82e8 − 3.33e8i)8-s + 9.75e8i·9-s + (−5.68e8 + 3.11e9i)10-s − 5.73e9i·11-s + (−5.93e9 − 4.01e9i)12-s + (−9.99e9 − 9.99e9i)13-s + (−4.04e9 − 4.24e10i)14-s + (5.74e10 + 1.62e10i)15-s + (−2.55e11 + 1.01e11i)16-s + (4.22e11 − 4.22e11i)17-s + ⋯ |
| L(s) = 1 | + (−0.770 − 0.636i)2-s + (−0.283 + 0.283i)3-s + (0.188 + 0.982i)4-s + (−0.487 − 0.872i)5-s + (0.399 − 0.0380i)6-s + (0.390 + 0.390i)7-s + (0.479 − 0.877i)8-s + 0.839i·9-s + (−0.179 + 0.983i)10-s − 0.733i·11-s + (−0.331 − 0.224i)12-s + (−0.261 − 0.261i)13-s + (−0.0523 − 0.549i)14-s + (0.385 + 0.109i)15-s + (−0.928 + 0.370i)16-s + (0.864 − 0.864i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.145i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.989 - 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(0.8281847499\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8281847499\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (558. + 461. i)T \) |
| 5 | \( 1 + (2.13e6 + 3.81e6i)T \) |
| good | 3 | \( 1 + (9.66e3 - 9.66e3i)T - 1.16e9iT^{2} \) |
| 7 | \( 1 + (-4.16e7 - 4.16e7i)T + 1.13e16iT^{2} \) |
| 11 | \( 1 + 5.73e9iT - 6.11e19T^{2} \) |
| 13 | \( 1 + (9.99e9 + 9.99e9i)T + 1.46e21iT^{2} \) |
| 17 | \( 1 + (-4.22e11 + 4.22e11i)T - 2.39e23iT^{2} \) |
| 19 | \( 1 + 1.82e12T + 1.97e24T^{2} \) |
| 23 | \( 1 + (1.02e13 - 1.02e13i)T - 7.46e25iT^{2} \) |
| 29 | \( 1 + 1.33e14iT - 6.10e27T^{2} \) |
| 31 | \( 1 - 2.44e14iT - 2.16e28T^{2} \) |
| 37 | \( 1 + (-7.05e13 + 7.05e13i)T - 6.24e29iT^{2} \) |
| 41 | \( 1 - 7.48e14T + 4.39e30T^{2} \) |
| 43 | \( 1 + (-1.66e15 + 1.66e15i)T - 1.08e31iT^{2} \) |
| 47 | \( 1 + (7.50e14 + 7.50e14i)T + 5.88e31iT^{2} \) |
| 53 | \( 1 + (-2.99e16 - 2.99e16i)T + 5.77e32iT^{2} \) |
| 59 | \( 1 - 5.34e16T + 4.42e33T^{2} \) |
| 61 | \( 1 - 9.54e16T + 8.34e33T^{2} \) |
| 67 | \( 1 + (-3.47e16 - 3.47e16i)T + 4.95e34iT^{2} \) |
| 71 | \( 1 - 2.20e17iT - 1.49e35T^{2} \) |
| 73 | \( 1 + (-3.79e17 - 3.79e17i)T + 2.53e35iT^{2} \) |
| 79 | \( 1 - 9.37e17T + 1.13e36T^{2} \) |
| 83 | \( 1 + (-4.82e16 + 4.82e16i)T - 2.90e36iT^{2} \) |
| 89 | \( 1 - 3.21e17iT - 1.09e37T^{2} \) |
| 97 | \( 1 + (7.00e18 - 7.00e18i)T - 5.60e37iT^{2} \) |
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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
−13.60574816211241085640725292339, −12.20199354971135890739888846851, −11.33249528353502867347683000246, −10.01644330529999009511635919734, −8.588752624797905593201177752861, −7.72952093053611192940805137256, −5.38974156183179722904072076847, −3.96860636566478329952081520957, −2.22566143273157834920834138718, −0.75029936052847113129337700115,
0.46100859079311121081109505473, 2.01249008860689855456159030683, 4.15143436722149298910368799802, 6.13448077650439988063173834262, 7.07715518461113547824508086773, 8.226035166218632396302394866707, 9.930886796144318240067813397920, 10.98056648617137309504935164310, 12.36972953354373408290376993679, 14.55068310880927170636335534569
