| L(s) = 1 | + (−0.571 + 0.421i)2-s + (0.0466 − 0.246i)3-s + (−0.440 + 1.42i)4-s + (2.23 − 0.0714i)5-s + (0.0773 + 0.160i)6-s + (0.120 − 2.64i)7-s + (−0.820 − 2.34i)8-s + (2.73 + 1.07i)9-s + (−1.24 + 0.983i)10-s + (0.529 + 1.34i)11-s + (0.331 + 0.175i)12-s + (0.370 + 3.28i)13-s + (1.04 + 1.56i)14-s + (0.0866 − 0.554i)15-s + (−1.01 − 0.690i)16-s + (0.0235 + 0.630i)17-s + ⋯ |
| L(s) = 1 | + (−0.404 + 0.298i)2-s + (0.0269 − 0.142i)3-s + (−0.220 + 0.714i)4-s + (0.999 − 0.0319i)5-s + (0.0315 + 0.0655i)6-s + (0.0456 − 0.998i)7-s + (−0.289 − 0.828i)8-s + (0.911 + 0.357i)9-s + (−0.394 + 0.311i)10-s + (0.159 + 0.406i)11-s + (0.0957 + 0.0505i)12-s + (0.102 + 0.910i)13-s + (0.279 + 0.417i)14-s + (0.0223 − 0.143i)15-s + (−0.253 − 0.172i)16-s + (0.00572 + 0.152i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 - 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.777 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.11987 + 0.396205i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11987 + 0.396205i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.23 + 0.0714i)T \) |
| 7 | \( 1 + (-0.120 + 2.64i)T \) |
| good | 2 | \( 1 + (0.571 - 0.421i)T + (0.589 - 1.91i)T^{2} \) |
| 3 | \( 1 + (-0.0466 + 0.246i)T + (-2.79 - 1.09i)T^{2} \) |
| 11 | \( 1 + (-0.529 - 1.34i)T + (-8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (-0.370 - 3.28i)T + (-12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (-0.0235 - 0.630i)T + (-16.9 + 1.27i)T^{2} \) |
| 19 | \( 1 + (-1.79 - 3.10i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.08 - 0.0779i)T + (22.9 + 1.71i)T^{2} \) |
| 29 | \( 1 + (-0.472 - 0.107i)T + (26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (5.64 + 3.25i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.69 + 6.99i)T + (-20.8 - 30.5i)T^{2} \) |
| 41 | \( 1 + (4.67 - 9.70i)T + (-25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (12.1 + 4.24i)T + (33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (1.13 + 1.53i)T + (-13.8 + 44.9i)T^{2} \) |
| 53 | \( 1 + (-0.557 - 1.05i)T + (-29.8 + 43.7i)T^{2} \) |
| 59 | \( 1 + (-0.818 + 10.9i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (3.36 + 10.9i)T + (-50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (1.31 + 4.90i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-1.90 - 8.35i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-0.424 + 0.574i)T + (-21.5 - 69.7i)T^{2} \) |
| 79 | \( 1 + (-8.89 + 5.13i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (11.1 + 1.25i)T + (80.9 + 18.4i)T^{2} \) |
| 89 | \( 1 + (3.16 - 8.07i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + (6.86 + 6.86i)T + 97iT^{2} \) |
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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
−12.44756675510073429300173146415, −11.12569328915743302365936479555, −9.921008573351043542740461850479, −9.484967825257450157262076943162, −8.181207866250674388010896726723, −7.20292281789909868902098269982, −6.54403876468515610388188077451, −4.80914718152197334153281078502, −3.69636037646540625160529902282, −1.69476739388159273240391403972,
1.41809005470475427086967059167, 2.89815007144622434700352687492, 4.98405220834240115290469771498, 5.71738363624491532374611910367, 6.80591589383038646810785997775, 8.534523329149626616026114363314, 9.255648360748878414011519325298, 9.989658916343209094343988755113, 10.76908942007634886745927674314, 11.86881530167431166349140791525
