| L(s) = 1 | + (0.190 + 0.129i)2-s + (−0.733 − 0.680i)3-s + (−0.711 − 1.81i)4-s + (1.13 − 0.350i)5-s + (−0.0512 − 0.224i)6-s + (−2.33 − 1.23i)7-s + (0.202 − 0.886i)8-s + (0.0747 + 0.997i)9-s + (0.261 + 0.0807i)10-s + (0.276 − 3.69i)11-s + (−0.711 + 1.81i)12-s + (1.00 − 0.485i)13-s + (−0.285 − 0.538i)14-s + (−1.07 − 0.516i)15-s + (−2.70 + 2.50i)16-s + (6.63 − 0.999i)17-s + ⋯ |
| L(s) = 1 | + (0.134 + 0.0917i)2-s + (−0.423 − 0.392i)3-s + (−0.355 − 0.906i)4-s + (0.508 − 0.156i)5-s + (−0.0209 − 0.0916i)6-s + (−0.884 − 0.466i)7-s + (0.0715 − 0.313i)8-s + (0.0249 + 0.332i)9-s + (0.0828 + 0.0255i)10-s + (0.0835 − 1.11i)11-s + (−0.205 + 0.523i)12-s + (0.279 − 0.134i)13-s + (−0.0761 − 0.143i)14-s + (−0.276 − 0.133i)15-s + (−0.675 + 0.626i)16-s + (1.60 − 0.242i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0866 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0866 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.707310 - 0.648449i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.707310 - 0.648449i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.733 + 0.680i)T \) |
| 7 | \( 1 + (2.33 + 1.23i)T \) |
| good | 2 | \( 1 + (-0.190 - 0.129i)T + (0.730 + 1.86i)T^{2} \) |
| 5 | \( 1 + (-1.13 + 0.350i)T + (4.13 - 2.81i)T^{2} \) |
| 11 | \( 1 + (-0.276 + 3.69i)T + (-10.8 - 1.63i)T^{2} \) |
| 13 | \( 1 + (-1.00 + 0.485i)T + (8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (-6.63 + 0.999i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (-2.49 - 4.32i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.09 + 0.164i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (-4.32 - 5.42i)T + (-6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (1.42 - 2.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.893 + 2.27i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (-0.970 + 4.25i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (2.16 + 9.48i)T + (-38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (-6.47 - 4.41i)T + (17.1 + 43.7i)T^{2} \) |
| 53 | \( 1 + (1.47 + 3.75i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-13.6 - 4.20i)T + (48.7 + 33.2i)T^{2} \) |
| 61 | \( 1 + (2.75 - 7.01i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (-1.69 + 2.94i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.49 - 4.37i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-1.17 + 0.798i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (8.37 + 14.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (12.1 + 5.84i)T + (51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (0.460 + 6.13i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + 2.94T + 97T^{2} \) |
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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
−12.99380384500005607049448867134, −11.92785790587964945375228782670, −10.53491903364057359052805542097, −9.982216158300971262555891797692, −8.807634043087187101369880005910, −7.27134392614272327548114106553, −5.95054843738869558717389276300, −5.51253578533724473473738386707, −3.56872500910439052279612022667, −1.08331349730686994066990232791,
2.80836876558035140131257714317, 4.18696866189173047132996276267, 5.55311917395345159611930294831, 6.79443605519759388367240109538, 8.087324685519776452617460790079, 9.549016465807205319434975265897, 9.904217025821454715462541095734, 11.56014359562972213939205994160, 12.30175139650382126973635136394, 13.11472603430426167222351922253
