| L(s) = 1 | + (0.173 − 0.984i)2-s + (−0.238 − 1.35i)3-s + (−0.939 − 0.342i)4-s + (0.266 − 0.223i)5-s − 1.37·6-s + (−0.365 + 0.307i)7-s + (−0.5 + 0.866i)8-s + (1.04 − 0.378i)9-s + (−0.173 − 0.300i)10-s + (−1.29 + 2.23i)11-s + (−0.238 + 1.35i)12-s + (4.21 + 1.53i)13-s + (0.238 + 0.413i)14-s + (−0.365 − 0.307i)15-s + (0.766 + 0.642i)16-s + (−1.88 + 0.687i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (−0.137 − 0.782i)3-s + (−0.469 − 0.171i)4-s + (0.118 − 0.0998i)5-s − 0.561·6-s + (−0.138 + 0.116i)7-s + (−0.176 + 0.306i)8-s + (0.346 − 0.126i)9-s + (−0.0549 − 0.0951i)10-s + (−0.389 + 0.675i)11-s + (−0.0689 + 0.391i)12-s + (1.16 + 0.425i)13-s + (0.0638 + 0.110i)14-s + (−0.0944 − 0.0792i)15-s + (0.191 + 0.160i)16-s + (−0.458 + 0.166i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0780 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0780 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.685204 - 0.633648i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.685204 - 0.633648i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 37 | \( 1 + (-2.42 + 5.57i)T \) |
| good | 3 | \( 1 + (0.238 + 1.35i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (-0.266 + 0.223i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (0.365 - 0.307i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (1.29 - 2.23i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.21 - 1.53i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (1.88 - 0.687i)T + (13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (-0.611 - 3.46i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (2.60 + 4.51i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.114 - 0.197i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 6.74T + 31T^{2} \) |
| 41 | \( 1 + (10.0 + 3.66i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 - 8.85T + 43T^{2} \) |
| 47 | \( 1 + (3.72 + 6.45i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.35 - 2.81i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-3.43 - 2.87i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.35 - 0.491i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (8.06 - 6.76i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.54 - 8.74i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 - 7.18T + 73T^{2} \) |
| 79 | \( 1 + (-2.70 + 2.27i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (1.32 - 0.483i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-2.92 - 2.45i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (9.24 + 16.0i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−13.95736931101161466563019878702, −12.98146702662578851877470989330, −12.36269609609318604853904845256, −11.15337420931369318327016728051, −9.977628361005391274526845600193, −8.700973526744253656010853392608, −7.23027869145878622455091587293, −5.85147533168094710322640210041, −4.01169869595108353775807037165, −1.82942286564869793950187444219,
3.64938138511806075628993087682, 5.10691019732322619661000002747, 6.38630541976562303954617090626, 7.899241766303637650717508146373, 9.150952604979135287796908089041, 10.32423843176886395918193050457, 11.33535989188602221652010839108, 13.09319224226518596399961019961, 13.74554880250188375278344279024, 15.15957799162391824025842500137
