| L(s) = 1 | + (0.101 + 0.703i)2-s + (−0.415 + 0.909i)3-s + (1.43 − 0.421i)4-s + (0.654 − 0.755i)5-s + (−0.681 − 0.200i)6-s + (−3.66 + 2.35i)7-s + (1.03 + 2.25i)8-s + (−0.654 − 0.755i)9-s + (0.597 + 0.384i)10-s + (−0.657 + 4.57i)11-s + (−0.212 + 1.47i)12-s + (1.03 + 0.665i)13-s + (−2.02 − 2.34i)14-s + (0.415 + 0.909i)15-s + (1.02 − 0.661i)16-s + (4.83 + 1.42i)17-s + ⋯ |
| L(s) = 1 | + (0.0715 + 0.497i)2-s + (−0.239 + 0.525i)3-s + (0.717 − 0.210i)4-s + (0.292 − 0.337i)5-s + (−0.278 − 0.0817i)6-s + (−1.38 + 0.890i)7-s + (0.364 + 0.798i)8-s + (−0.218 − 0.251i)9-s + (0.189 + 0.121i)10-s + (−0.198 + 1.37i)11-s + (−0.0614 + 0.427i)12-s + (0.287 + 0.184i)13-s + (−0.542 − 0.625i)14-s + (0.107 + 0.234i)15-s + (0.257 − 0.165i)16-s + (1.17 + 0.344i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.191 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.191 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.887629 + 1.07795i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.887629 + 1.07795i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.415 - 0.909i)T \) |
| 5 | \( 1 + (-0.654 + 0.755i)T \) |
| 23 | \( 1 + (-1.09 - 4.66i)T \) |
| good | 2 | \( 1 + (-0.101 - 0.703i)T + (-1.91 + 0.563i)T^{2} \) |
| 7 | \( 1 + (3.66 - 2.35i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.657 - 4.57i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-1.03 - 0.665i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-4.83 - 1.42i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-2.46 + 0.723i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (1.56 + 0.459i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (3.89 + 8.52i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (5.06 + 5.84i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-3.94 + 4.54i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (0.578 - 1.26i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 7.40T + 47T^{2} \) |
| 53 | \( 1 + (-3.96 + 2.55i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-7.21 - 4.63i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (0.252 + 0.552i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (1.71 + 11.9i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (1.09 + 7.60i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-11.7 + 3.44i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (-11.7 - 7.57i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (1.66 + 1.92i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-0.0938 + 0.205i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (11.0 - 12.7i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.92108567795096872503773482881, −10.74622287393438420207762151625, −9.668389438627078551349886848486, −9.384064435554029143739190845450, −7.81534193244975165509953484111, −6.83621989589926607021298268289, −5.81224134380999279987376680166, −5.27449685756708779039361174119, −3.56254801326867864219909978019, −2.13832833043970681714915258927,
1.01840117994054077614795709006, 2.99452014969700846139783174547, 3.47934777012932054205540848702, 5.61744411804056328401969638347, 6.57543473302380711473253965110, 7.15495704272653594521802418814, 8.313642148514488611201679026479, 9.797330794240419590649064703622, 10.48618086173543165017475539598, 11.17770188693362433825390185204
