| 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
−12.24600981371975163616707700181, −10.36278656787255326429944194187, −9.611780781904429337240750339973, −9.194862140780381366449929034408, −8.021648956390971624985421622113, −7.44803334022533381597023320144, −6.40066578341099050255456309787, −5.68449605764464331072491244759, −3.99088318837844890227330552603, −1.75549160069435771815232359319,
0.41343480031724645052599778237, 2.62232059259447740633086649081, 3.20576188956783717305151671013, 4.42380049785367495948975873258, 6.74481392314033690335885674784, 7.48618867587525539328623733388, 8.866341301063068156928201367966, 9.340198929595902832758816559636, 9.991518052551317741033085553308, 10.99225904956579771747784542924
