| L(s) = 1 | + (−0.766 + 0.642i)2-s + (1.57 − 0.715i)3-s + (0.173 − 0.984i)4-s + (0.262 − 0.0462i)5-s + (−0.748 + 1.56i)6-s + (0.604 − 1.04i)7-s + (0.500 + 0.866i)8-s + (1.97 − 2.25i)9-s + (−0.171 + 0.204i)10-s + (−2.03 + 1.17i)11-s + (−0.431 − 1.67i)12-s + (1.01 + 2.79i)13-s + (0.209 + 1.19i)14-s + (0.380 − 0.260i)15-s + (−0.939 − 0.342i)16-s + (−0.576 − 0.687i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 + 0.454i)2-s + (0.910 − 0.413i)3-s + (0.0868 − 0.492i)4-s + (0.117 − 0.0206i)5-s + (−0.305 + 0.637i)6-s + (0.228 − 0.395i)7-s + (0.176 + 0.306i)8-s + (0.658 − 0.752i)9-s + (−0.0541 + 0.0645i)10-s + (−0.613 + 0.354i)11-s + (−0.124 − 0.484i)12-s + (0.282 + 0.775i)13-s + (0.0561 + 0.318i)14-s + (0.0982 − 0.0673i)15-s + (−0.234 − 0.0855i)16-s + (−0.139 − 0.166i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.06692 + 0.0128935i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06692 + 0.0128935i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 3 | \( 1 + (-1.57 + 0.715i)T \) |
| 19 | \( 1 + (-1.97 - 3.88i)T \) |
| good | 5 | \( 1 + (-0.262 + 0.0462i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.604 + 1.04i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.03 - 1.17i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.01 - 2.79i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (0.576 + 0.687i)T + (-2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (5.53 + 0.976i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (1.92 + 1.61i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (8.98 + 5.18i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 3.95iT - 37T^{2} \) |
| 41 | \( 1 + (-10.4 - 3.79i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.834 + 4.73i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (1.24 - 1.48i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (0.998 - 5.66i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-9.78 + 8.20i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (0.153 - 0.871i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.28 + 3.91i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.64 - 9.33i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-0.320 - 0.116i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-2.33 + 6.42i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (12.2 + 7.05i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-11.5 + 4.20i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-3.88 - 4.62i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−13.84422615256430038716175106879, −12.79668619264856808143978261602, −11.46018944105724170961199864962, −10.05346954081318253380467791551, −9.258300367719379830078048689807, −7.994121717789706914512428924240, −7.35808614611927033180319035084, −5.95252199854170901972792025127, −4.04347636289365385774522841437, −1.95500344041229009937522224334,
2.30814860113295529854540136606, 3.69866856239145669790281304119, 5.45156786676626132512074077102, 7.46294068282164518343701569017, 8.379799944519660073195976736612, 9.301513464787916902509922604763, 10.33478697596032712243500157653, 11.22723815833138819813335882005, 12.65097043707213186479914021116, 13.51834706136915056364000517282
