| L(s) = 1 | + (−1.10 − 1.27i)2-s + (0.372 − 0.815i)3-s + (−0.118 + 0.822i)4-s + (1.36 + 0.402i)5-s + (−1.44 + 0.424i)6-s + (−3.35 − 3.86i)7-s + (−1.65 + 1.06i)8-s + (1.43 + 1.66i)9-s + (−0.997 − 2.18i)10-s + (3.02 + 0.887i)11-s + (0.626 + 0.402i)12-s + (2.17 + 1.40i)13-s + (−1.22 + 8.51i)14-s + (0.837 − 0.966i)15-s + (4.77 + 1.40i)16-s + (0.327 + 2.27i)17-s + ⋯ |
| L(s) = 1 | + (−0.779 − 0.899i)2-s + (0.214 − 0.470i)3-s + (−0.0591 + 0.411i)4-s + (0.612 + 0.179i)5-s + (−0.590 + 0.173i)6-s + (−1.26 − 1.46i)7-s + (−0.585 + 0.376i)8-s + (0.479 + 0.553i)9-s + (−0.315 − 0.690i)10-s + (0.911 + 0.267i)11-s + (0.180 + 0.116i)12-s + (0.604 + 0.388i)13-s + (−0.327 + 2.27i)14-s + (0.216 − 0.249i)15-s + (1.19 + 0.350i)16-s + (0.0794 + 0.552i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.431232 - 0.536621i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.431232 - 0.536621i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 67 | \( 1 + (-8.09 - 1.21i)T \) |
| good | 2 | \( 1 + (1.10 + 1.27i)T + (-0.284 + 1.97i)T^{2} \) |
| 3 | \( 1 + (-0.372 + 0.815i)T + (-1.96 - 2.26i)T^{2} \) |
| 5 | \( 1 + (-1.36 - 0.402i)T + (4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (3.35 + 3.86i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-3.02 - 0.887i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-2.17 - 1.40i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-0.327 - 2.27i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-1.98 + 2.29i)T + (-2.70 - 18.8i)T^{2} \) |
| 23 | \( 1 + (3.12 - 6.84i)T + (-15.0 - 17.3i)T^{2} \) |
| 29 | \( 1 - 0.448T + 29T^{2} \) |
| 31 | \( 1 + (2.78 - 1.79i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + 3.87T + 37T^{2} \) |
| 41 | \( 1 + (1.15 + 8.06i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (0.217 + 1.51i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + (0.776 - 1.70i)T + (-30.7 - 35.5i)T^{2} \) |
| 53 | \( 1 + (-0.184 + 1.28i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (10.3 - 6.67i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (7.48 - 2.19i)T + (51.3 - 32.9i)T^{2} \) |
| 71 | \( 1 + (-1.78 + 12.4i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-5.56 + 1.63i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (9.94 + 6.39i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (0.440 + 0.129i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-4.16 - 9.12i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−13.99343943912259072758408398149, −13.47913877991112001054411396286, −12.22030661688648754297341089613, −10.79207037332369214199801840312, −10.02168805975107314896281063907, −9.191044688556440899531799323521, −7.42825765489381969932567043157, −6.27789885319340798948732210323, −3.64797269983527650018884757634, −1.59301527467922879782683336589,
3.34308881727280948830198198981, 5.88019067932742927496668731402, 6.59250245829585886719302614259, 8.475417146086133563316196118287, 9.350980970125799643335498877292, 9.834813794247567899325439396475, 12.02324583120488172029908131679, 12.85670845111654034593082358998, 14.47288308727787546834565572910, 15.57399038543203948180637718117
