| L(s) = 1 | + (−0.758 + 0.875i)2-s + (−0.841 + 0.540i)3-s + (0.0936 + 0.651i)4-s + (−0.415 − 0.909i)5-s + (0.164 − 1.14i)6-s + (−1.51 + 0.446i)7-s + (−2.59 − 1.66i)8-s + (0.415 − 0.909i)9-s + (1.11 + 0.326i)10-s + (−0.942 − 1.08i)11-s + (−0.431 − 0.497i)12-s + (−2.71 − 0.798i)13-s + (0.762 − 1.66i)14-s + (0.841 + 0.540i)15-s + (2.15 − 0.634i)16-s + (0.357 − 2.48i)17-s + ⋯ |
| L(s) = 1 | + (−0.536 + 0.619i)2-s + (−0.485 + 0.312i)3-s + (0.0468 + 0.325i)4-s + (−0.185 − 0.406i)5-s + (0.0673 − 0.468i)6-s + (−0.574 + 0.168i)7-s + (−0.915 − 0.588i)8-s + (0.138 − 0.303i)9-s + (0.351 + 0.103i)10-s + (−0.284 − 0.327i)11-s + (−0.124 − 0.143i)12-s + (−0.754 − 0.221i)13-s + (0.203 − 0.445i)14-s + (0.217 + 0.139i)15-s + (0.539 − 0.158i)16-s + (0.0868 − 0.603i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0679 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0679 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0940572 - 0.100678i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0940572 - 0.100678i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (3.88 + 2.81i)T \) |
| good | 2 | \( 1 + (0.758 - 0.875i)T + (-0.284 - 1.97i)T^{2} \) |
| 7 | \( 1 + (1.51 - 0.446i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (0.942 + 1.08i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (2.71 + 0.798i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.357 + 2.48i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.236 - 1.64i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.710 + 4.94i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (0.239 + 0.154i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-1.36 + 2.98i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (1.55 + 3.40i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (5.20 - 3.34i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 3.54T + 47T^{2} \) |
| 53 | \( 1 + (10.0 - 2.94i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (6.13 + 1.80i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-1.80 - 1.15i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (3.78 - 4.36i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (5.60 - 6.46i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.0961 - 0.668i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-7.09 - 2.08i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (6.97 - 15.2i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (5.70 - 3.66i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (4.51 + 9.88i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−11.34146184373421104265145478142, −10.03204321771863492941990882920, −9.427029819154081952603382811912, −8.363542024467706571094736020886, −7.56216931617513357984426278504, −6.49526759344747364321678043097, −5.57379727686666135514499493021, −4.24173771376975000771037718547, −2.90893776557932691218708280912, −0.11451764349704306005241862261,
1.80075424659828935068365803043, 3.17336686162775252130976807818, 4.87184241306833410388346637322, 6.05625214400862901927042560452, 6.91940925705802471804314700242, 8.016106047657094442310720578203, 9.304268038435602655666546922160, 10.08820278904008320122377843439, 10.71193858369005894570757091500, 11.66546020418764459186836424970
