| L(s) = 1 | + (0.875 + 0.0655i)2-s + (1.63 − 0.563i)3-s + (−1.21 − 0.183i)4-s + (0.193 − 0.848i)5-s + (1.47 − 0.386i)6-s + (2.18 − 1.49i)7-s + (−2.76 − 0.630i)8-s + (2.36 − 1.84i)9-s + (0.225 − 0.729i)10-s + (−0.798 − 1.65i)11-s + (−2.09 + 0.385i)12-s + (0.536 + 0.0401i)13-s + (2.00 − 1.16i)14-s + (−0.161 − 1.49i)15-s + (−0.0270 − 0.00833i)16-s + (−0.957 + 0.144i)17-s + ⋯ |
| L(s) = 1 | + (0.618 + 0.0463i)2-s + (0.945 − 0.325i)3-s + (−0.608 − 0.0916i)4-s + (0.0866 − 0.379i)5-s + (0.600 − 0.157i)6-s + (0.825 − 0.564i)7-s + (−0.977 − 0.223i)8-s + (0.787 − 0.615i)9-s + (0.0712 − 0.230i)10-s + (−0.240 − 0.500i)11-s + (−0.604 + 0.111i)12-s + (0.148 + 0.0111i)13-s + (0.536 − 0.311i)14-s + (−0.0416 − 0.387i)15-s + (−0.00675 − 0.00208i)16-s + (−0.232 + 0.0350i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.02294 - 1.03176i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.02294 - 1.03176i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.63 + 0.563i)T \) |
| 7 | \( 1 + (-2.18 + 1.49i)T \) |
| good | 2 | \( 1 + (-0.875 - 0.0655i)T + (1.97 + 0.298i)T^{2} \) |
| 5 | \( 1 + (-0.193 + 0.848i)T + (-4.50 - 2.16i)T^{2} \) |
| 11 | \( 1 + (0.798 + 1.65i)T + (-6.85 + 8.60i)T^{2} \) |
| 13 | \( 1 + (-0.536 - 0.0401i)T + (12.8 + 1.93i)T^{2} \) |
| 17 | \( 1 + (0.957 - 0.144i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (2.82 - 1.62i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.99 - 3.18i)T + (5.11 + 22.4i)T^{2} \) |
| 29 | \( 1 + (2.42 + 0.952i)T + (21.2 + 19.7i)T^{2} \) |
| 31 | \( 1 + (-3.04 + 1.75i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.43 - 3.66i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (-2.25 - 2.08i)T + (3.06 + 40.8i)T^{2} \) |
| 43 | \( 1 + (3.25 - 3.01i)T + (3.21 - 42.8i)T^{2} \) |
| 47 | \( 1 + (-0.0375 + 0.500i)T + (-46.4 - 7.00i)T^{2} \) |
| 53 | \( 1 + (10.0 - 3.95i)T + (38.8 - 36.0i)T^{2} \) |
| 59 | \( 1 + (7.02 - 6.51i)T + (4.40 - 58.8i)T^{2} \) |
| 61 | \( 1 + (-1.22 - 8.14i)T + (-58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (-4.48 - 7.77i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (11.1 + 8.87i)T + (15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-4.26 - 6.26i)T + (-26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + (1.55 - 2.69i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.0843 + 1.12i)T + (-82.0 + 12.3i)T^{2} \) |
| 89 | \( 1 + (0.00186 + 0.0248i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + (-12.7 + 7.36i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.04236830601095691395222147047, −9.937419892641774918218901967857, −8.951307863896994923657647592630, −8.380548485672865679216132721822, −7.43834549451628804104631255302, −6.18777436043591511765648584916, −4.95614202037125698267055947346, −4.14049504759902442555801836499, −3.04003639058123133715775192727, −1.28255967713563250960105458993,
2.21315900621809977505097022020, 3.25877736656745029088392995762, 4.53041803839602383954847029619, 5.06781500457620310306600442319, 6.54408560864848350730798931613, 7.80279978500601038908300798097, 8.680358166076649649509266625464, 9.206283506178550041593039372626, 10.36035261575186749569546010040, 11.20540935661573993663905466246
