L(s) = 1 | + (−0.309 − 0.951i)2-s + (0.809 − 1.53i)3-s + (−0.809 + 0.587i)4-s + (−0.897 − 0.291i)5-s + (−1.70 − 0.296i)6-s + (0.897 + 1.23i)7-s + (0.809 + 0.587i)8-s + (−1.69 − 2.47i)9-s + 0.943i·10-s + (0.151 + 3.31i)11-s + (0.245 + 1.71i)12-s + (4.03 − 1.30i)13-s + (0.897 − 1.23i)14-s + (−1.17 + 1.13i)15-s + (0.309 − 0.951i)16-s + (−0.906 + 2.78i)17-s + ⋯ |
L(s) = 1 | + (−0.218 − 0.672i)2-s + (0.467 − 0.884i)3-s + (−0.404 + 0.293i)4-s + (−0.401 − 0.130i)5-s + (−0.696 − 0.120i)6-s + (0.339 + 0.466i)7-s + (0.286 + 0.207i)8-s + (−0.563 − 0.826i)9-s + 0.298i·10-s + (0.0457 + 0.998i)11-s + (0.0709 + 0.494i)12-s + (1.11 − 0.363i)13-s + (0.239 − 0.330i)14-s + (−0.302 + 0.293i)15-s + (0.0772 − 0.237i)16-s + (−0.219 + 0.676i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.208 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.208 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.674204 - 0.545586i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.674204 - 0.545586i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 3 | \( 1 + (-0.809 + 1.53i)T \) |
| 11 | \( 1 + (-0.151 - 3.31i)T \) |
good | 5 | \( 1 + (0.897 + 0.291i)T + (4.04 + 2.93i)T^{2} \) |
| 7 | \( 1 + (-0.897 - 1.23i)T + (-2.16 + 6.65i)T^{2} \) |
| 13 | \( 1 + (-4.03 + 1.30i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (0.906 - 2.78i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (4.16 - 5.73i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + 1.18iT - 23T^{2} \) |
| 29 | \( 1 + (-5.17 + 3.75i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (2.68 + 8.27i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (2.28 - 1.66i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (5.92 + 4.30i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 5.34iT - 43T^{2} \) |
| 47 | \( 1 + (5.58 - 7.68i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-4.62 + 1.50i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-2.74 - 3.77i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (0.685 + 0.222i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 - 3.89T + 67T^{2} \) |
| 71 | \( 1 + (-9.47 - 3.07i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-1.03 - 1.41i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (1.56 - 0.508i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-1.90 + 5.85i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + 17.3iT - 89T^{2} \) |
| 97 | \( 1 + (-3.54 - 10.8i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−14.51797524295338137100096337014, −13.21888908362595909074040530215, −12.41352586767601314524273813910, −11.53455480979764724732602494211, −10.09336666659811382626997051098, −8.578181158405468166898695301466, −7.933884991667349642889561144066, −6.18434389903863806633736044042, −3.96031595649973065902395205791, −1.98224273110715892290112230774,
3.62268449802275999316575604826, 5.05367016906947677232204408446, 6.80294551433325624591476056817, 8.336484245552180820878133796876, 9.017382449223432388362948263542, 10.59168563274683766774507700903, 11.33406210937897771645473841047, 13.49586319838369217425975750104, 14.10015636477508570396634747002, 15.33829659690734846205646565275