| L(s) = 1 | + (−0.775 − 1.18i)2-s + (0.720 + 1.57i)3-s + (−0.797 + 1.83i)4-s + (−0.992 − 1.13i)5-s + (1.30 − 2.07i)6-s + (−1.51 − 0.199i)7-s + (2.78 − 0.478i)8-s + (−1.96 + 2.27i)9-s + (−0.569 + 2.05i)10-s + (−1.85 − 3.75i)11-s + (−3.46 + 0.0662i)12-s + (0.737 + 2.17i)13-s + (0.940 + 1.94i)14-s + (1.06 − 2.37i)15-s + (−2.72 − 2.92i)16-s + (−4.20 − 4.20i)17-s + ⋯ |
| L(s) = 1 | + (−0.548 − 0.836i)2-s + (0.416 + 0.909i)3-s + (−0.398 + 0.917i)4-s + (−0.443 − 0.506i)5-s + (0.532 − 0.846i)6-s + (−0.573 − 0.0754i)7-s + (0.985 − 0.169i)8-s + (−0.653 + 0.756i)9-s + (−0.179 + 0.648i)10-s + (−0.558 − 1.13i)11-s + (−0.999 + 0.0191i)12-s + (0.204 + 0.602i)13-s + (0.251 + 0.520i)14-s + (0.275 − 0.614i)15-s + (−0.681 − 0.731i)16-s + (−1.01 − 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0719180 - 0.352482i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0719180 - 0.352482i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.775 + 1.18i)T \) |
| 3 | \( 1 + (-0.720 - 1.57i)T \) |
| good | 5 | \( 1 + (0.992 + 1.13i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (1.51 + 0.199i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (1.85 + 3.75i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (-0.737 - 2.17i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (4.20 + 4.20i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.0676 + 0.340i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (-0.115 - 0.876i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (-1.53 + 0.100i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (3.11 - 1.79i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.96 + 9.90i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (1.45 + 11.0i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (6.89 - 3.40i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (-5.41 + 1.45i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (1.12 - 1.67i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (2.94 + 3.35i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (-0.340 - 5.19i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (8.96 + 4.42i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (-5.31 - 12.8i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (2.93 - 7.08i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-3.55 - 13.2i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-1.19 - 1.04i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (-2.83 + 1.17i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (-11.6 - 6.72i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.40590258324541115024438056676, −9.330602367735037912767338370487, −8.867019542734857986881689593888, −8.158374908179452677541394677735, −7.00222176205414475329337395163, −5.37082246813143369613278130769, −4.30907430776334652429891515097, −3.47630113545144642723574521148, −2.43225953923241086407234280133, −0.22206521563127396364694886677,
1.79136238063192540568198619013, 3.23772377034886090549191889531, 4.74753888689237675283166235191, 6.14006952312216845311078561855, 6.72475711912108859974211816947, 7.58353455635506419940446025388, 8.201982505307263564710779242417, 9.132541716937979231895130394538, 10.08284079249278859432597533068, 10.88390417246618418139104044546
