| L(s) = 1 | + (−0.580 − 1.28i)2-s + (0.342 − 0.939i)3-s + (−1.32 + 1.49i)4-s + (0.853 − 0.150i)5-s + (−1.41 + 0.104i)6-s + (0.0398 − 0.0690i)7-s + (2.70 + 0.841i)8-s + (−0.766 − 0.642i)9-s + (−0.689 − 1.01i)10-s + (0.739 − 0.427i)11-s + (0.953 + 1.75i)12-s + (−2.18 − 6.01i)13-s + (−0.112 − 0.0113i)14-s + (0.150 − 0.853i)15-s + (−0.482 − 3.97i)16-s + (0.464 − 0.389i)17-s + ⋯ |
| L(s) = 1 | + (−0.410 − 0.911i)2-s + (0.197 − 0.542i)3-s + (−0.663 + 0.748i)4-s + (0.381 − 0.0673i)5-s + (−0.575 + 0.0426i)6-s + (0.0150 − 0.0260i)7-s + (0.954 + 0.297i)8-s + (−0.255 − 0.214i)9-s + (−0.218 − 0.320i)10-s + (0.223 − 0.128i)11-s + (0.275 + 0.507i)12-s + (−0.607 − 1.66i)13-s + (−0.0299 − 0.00302i)14-s + (0.0388 − 0.220i)15-s + (−0.120 − 0.992i)16-s + (0.112 − 0.0944i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.785 + 0.618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.785 + 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.358381 - 1.03518i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.358381 - 1.03518i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.580 + 1.28i)T \) |
| 3 | \( 1 + (-0.342 + 0.939i)T \) |
| 19 | \( 1 + (-3.00 + 3.15i)T \) |
| good | 5 | \( 1 + (-0.853 + 0.150i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.0398 + 0.0690i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.739 + 0.427i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.18 + 6.01i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.464 + 0.389i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.655 + 3.71i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (4.15 - 4.94i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-5.02 + 8.69i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 0.193iT - 37T^{2} \) |
| 41 | \( 1 + (-5.97 - 2.17i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (3.40 - 0.600i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-2.84 - 2.39i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (10.2 + 1.80i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-6.74 - 8.03i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (8.93 + 1.57i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-7.25 + 8.64i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (0.0904 + 0.512i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-3.99 - 1.45i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (0.598 + 0.217i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (1.07 + 0.618i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-12.6 + 4.58i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-3.79 + 3.18i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.73360193229758601806835869217, −9.793714763586417330193228811622, −9.124041258825026196463499388094, −7.983534615654755591337427217680, −7.44169542258747259523556081180, −5.94391800663257190049370137193, −4.81098787176177745397508711665, −3.30737978028174624134131408912, −2.37095603522004485056495617626, −0.796633249365987755872732942292,
1.85401904507151925909711528745, 3.83503904169741884780205767084, 4.85441994932480600485069098273, 5.86536348982409693217159291153, 6.84960456272114739586184475808, 7.76201550184192721938098487260, 8.819833517473696362278597709592, 9.621300978748783204931425304591, 10.02363844760848092622956333715, 11.27490520534102218421067957633
