| L(s) = 1 | + (−1.71 − 1.97i)2-s + (0.841 + 0.540i)3-s + (−0.688 + 4.78i)4-s + (−0.415 + 0.909i)5-s + (−0.372 − 2.58i)6-s + (−3.22 − 0.946i)7-s + (6.24 − 4.01i)8-s + (0.415 + 0.909i)9-s + (2.50 − 0.736i)10-s + (2.97 − 3.43i)11-s + (−3.16 + 3.65i)12-s + (−4.24 + 1.24i)13-s + (3.64 + 7.98i)14-s + (−0.841 + 0.540i)15-s + (−9.33 − 2.74i)16-s + (−0.923 − 6.42i)17-s + ⋯ |
| L(s) = 1 | + (−1.21 − 1.39i)2-s + (0.485 + 0.312i)3-s + (−0.344 + 2.39i)4-s + (−0.185 + 0.406i)5-s + (−0.151 − 1.05i)6-s + (−1.21 − 0.357i)7-s + (2.20 − 1.41i)8-s + (0.138 + 0.303i)9-s + (0.793 − 0.232i)10-s + (0.897 − 1.03i)11-s + (−0.914 + 1.05i)12-s + (−1.17 + 0.346i)13-s + (0.974 + 2.13i)14-s + (−0.217 + 0.139i)15-s + (−2.33 − 0.685i)16-s + (−0.224 − 1.55i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0300423 - 0.431050i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0300423 - 0.431050i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.841 - 0.540i)T \) |
| 5 | \( 1 + (0.415 - 0.909i)T \) |
| 23 | \( 1 + (3.08 + 3.67i)T \) |
| good | 2 | \( 1 + (1.71 + 1.97i)T + (-0.284 + 1.97i)T^{2} \) |
| 7 | \( 1 + (3.22 + 0.946i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (-2.97 + 3.43i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (4.24 - 1.24i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (0.923 + 6.42i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-1.07 + 7.50i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.103 - 0.718i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (2.89 - 1.86i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-1.61 - 3.53i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (-0.199 + 0.437i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-2.82 - 1.81i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + (10.9 + 3.20i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (5.83 - 1.71i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (1.15 - 0.740i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-5.68 - 6.56i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (3.41 + 3.94i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.691 + 4.80i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (4.55 - 1.33i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-0.0538 - 0.118i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (1.88 + 1.20i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-2.93 + 6.42i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.99225904956579771747784542924, −9.991518052551317741033085553308, −9.340198929595902832758816559636, −8.866341301063068156928201367966, −7.48618867587525539328623733388, −6.74481392314033690335885674784, −4.42380049785367495948975873258, −3.20576188956783717305151671013, −2.62232059259447740633086649081, −0.41343480031724645052599778237,
1.75549160069435771815232359319, 3.99088318837844890227330552603, 5.68449605764464331072491244759, 6.40066578341099050255456309787, 7.44803334022533381597023320144, 8.021648956390971624985421622113, 9.194862140780381366449929034408, 9.611780781904429337240750339973, 10.36278656787255326429944194187, 12.24600981371975163616707700181
