| 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
−11.17770188693362433825390185204, −10.48618086173543165017475539598, −9.797330794240419590649064703622, −8.313642148514488611201679026479, −7.15495704272653594521802418814, −6.57543473302380711473253965110, −5.61744411804056328401969638347, −3.47934777012932054205540848702, −2.99452014969700846139783174547, −1.01840117994054077614795709006,
2.13832833043970681714915258927, 3.56254801326867864219909978019, 5.27449685756708779039361174119, 5.81224134380999279987376680166, 6.83621989589926607021298268289, 7.81534193244975165509953484111, 9.384064435554029143739190845450, 9.668389438627078551349886848486, 10.74622287393438420207762151625, 11.92108567795096872503773482881
