| L(s) = 1 | + (−0.0935 + 1.78i)2-s + (0.279 − 0.226i)3-s + (−1.19 − 0.125i)4-s + (1.07 − 1.95i)5-s + (0.378 + 0.520i)6-s + (−0.314 + 2.62i)7-s + (−0.224 + 1.41i)8-s + (−0.596 + 2.80i)9-s + (3.39 + 2.11i)10-s + (4.12 − 0.877i)11-s + (−0.361 + 0.234i)12-s + (0.268 − 0.527i)13-s + (−4.66 − 0.808i)14-s + (−0.141 − 0.792i)15-s + (−4.85 − 1.03i)16-s + (−0.811 + 0.311i)17-s + ⋯ |
| L(s) = 1 | + (−0.0661 + 1.26i)2-s + (0.161 − 0.130i)3-s + (−0.595 − 0.0626i)4-s + (0.482 − 0.875i)5-s + (0.154 + 0.212i)6-s + (−0.119 + 0.992i)7-s + (−0.0793 + 0.500i)8-s + (−0.198 + 0.935i)9-s + (1.07 + 0.667i)10-s + (1.24 − 0.264i)11-s + (−0.104 + 0.0677i)12-s + (0.0745 − 0.146i)13-s + (−1.24 − 0.215i)14-s + (−0.0365 − 0.204i)15-s + (−1.21 − 0.257i)16-s + (−0.196 + 0.0755i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0443 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0443 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.893362 + 0.933935i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.893362 + 0.933935i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.07 + 1.95i)T \) |
| 7 | \( 1 + (0.314 - 2.62i)T \) |
| good | 2 | \( 1 + (0.0935 - 1.78i)T + (-1.98 - 0.209i)T^{2} \) |
| 3 | \( 1 + (-0.279 + 0.226i)T + (0.623 - 2.93i)T^{2} \) |
| 11 | \( 1 + (-4.12 + 0.877i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.268 + 0.527i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (0.811 - 0.311i)T + (12.6 - 11.3i)T^{2} \) |
| 19 | \( 1 + (0.667 + 6.34i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (1.61 + 0.0848i)T + (22.8 + 2.40i)T^{2} \) |
| 29 | \( 1 + (0.996 - 1.37i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-3.65 + 8.22i)T + (-20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (2.37 + 3.65i)T + (-15.0 + 33.8i)T^{2} \) |
| 41 | \( 1 + (5.70 - 1.85i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-3.73 + 3.73i)T - 43iT^{2} \) |
| 47 | \( 1 + (3.45 - 8.99i)T + (-34.9 - 31.4i)T^{2} \) |
| 53 | \( 1 + (-5.33 - 6.59i)T + (-11.0 + 51.8i)T^{2} \) |
| 59 | \( 1 + (-4.53 + 5.03i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (0.917 - 0.826i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (4.10 + 10.7i)T + (-49.7 + 44.8i)T^{2} \) |
| 71 | \( 1 + (11.3 + 8.23i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.92 - 1.89i)T + (29.6 + 66.6i)T^{2} \) |
| 79 | \( 1 + (-5.74 - 12.9i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-3.01 - 0.477i)T + (78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (-11.4 - 12.6i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (11.5 - 1.83i)T + (92.2 - 29.9i)T^{2} \) |
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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
−13.30324502963263853410533236531, −12.08343926801369146813089013471, −11.12832837418520788470546843633, −9.339507226615763152392983851746, −8.790865271191678078506469563370, −7.903879914395112911169796553397, −6.54220747236141164143369944178, −5.69368916027785546912160876926, −4.69987701092468526192099556245, −2.27780110539744026835555132910,
1.55934790217064757477056199825, 3.30206069371812071122280297726, 4.01261488633731249284579447765, 6.32663613869579827448238088694, 7.00618883127546318871016436728, 8.833758903773464827714388731479, 10.02972202865390136779398068709, 10.23545654822372905978165588560, 11.54728698347327874271219772410, 12.09945458666828680046465920657
