| L(s) = 1 | + (1.41 − 0.0137i)2-s + (−1.25 − 0.857i)3-s + (1.99 − 0.0389i)4-s + (−0.263 − 0.0197i)5-s + (−1.79 − 1.19i)6-s + (2.03 − 1.68i)7-s + (2.82 − 0.0825i)8-s + (−0.248 − 0.633i)9-s + (−0.372 − 0.0242i)10-s + (1.57 + 0.619i)11-s + (−2.54 − 1.66i)12-s + (−2.83 + 2.26i)13-s + (2.85 − 2.41i)14-s + (0.314 + 0.250i)15-s + (3.99 − 0.155i)16-s + (0.766 + 0.826i)17-s + ⋯ |
| L(s) = 1 | + (0.999 − 0.00972i)2-s + (−0.726 − 0.495i)3-s + (0.999 − 0.0194i)4-s + (−0.117 − 0.00883i)5-s + (−0.731 − 0.488i)6-s + (0.770 − 0.637i)7-s + (0.999 − 0.0291i)8-s + (−0.0828 − 0.211i)9-s + (−0.117 − 0.00768i)10-s + (0.475 + 0.186i)11-s + (−0.735 − 0.481i)12-s + (−0.787 + 0.628i)13-s + (0.763 − 0.645i)14-s + (0.0812 + 0.0647i)15-s + (0.999 − 0.0388i)16-s + (0.186 + 0.200i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 + 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.67831 - 0.554227i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.67831 - 0.554227i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.41 + 0.0137i)T \) |
| 7 | \( 1 + (-2.03 + 1.68i)T \) |
| good | 3 | \( 1 + (1.25 + 0.857i)T + (1.09 + 2.79i)T^{2} \) |
| 5 | \( 1 + (0.263 + 0.0197i)T + (4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (-1.57 - 0.619i)T + (8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (2.83 - 2.26i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-0.766 - 0.826i)T + (-1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (0.734 - 1.27i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.16 - 2.33i)T + (-1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-1.25 - 5.50i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (2.89 + 5.01i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.55 + 0.788i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (-2.59 - 5.39i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (4.36 - 9.06i)T + (-26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (-5.64 + 0.851i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (0.0663 - 0.0204i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (0.838 + 11.1i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (-0.0587 + 0.190i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-12.8 + 7.43i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.45 + 0.788i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-0.419 + 2.78i)T + (-69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (13.0 + 7.52i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.20 - 4.01i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-16.7 + 6.56i)T + (65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + 7.85iT - 97T^{2} \) |
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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
−12.26530839800180729952928189373, −11.67591445901848679766738324549, −10.94665506464646280797736072819, −9.675516573925855016391739636517, −7.892004560564286642705815145865, −7.01485621212233235943022004636, −6.06415676860504111058788880911, −4.88967554656995770124851858654, −3.77390339657578376133110197725, −1.69686952948166534093824498232,
2.37183847804579476888997565712, 4.12661226285264636668504785855, 5.19798891604967193192832857292, 5.83782336739162408668090155518, 7.29027063169241799815574789817, 8.424039788445181196679389091731, 10.05711577989969179098641768090, 10.92155781465305789222220389165, 11.80670788602577542640685678528, 12.28160573596482731564429811483
