| L(s) = 1 | + (0.959 + 0.281i)2-s + (−1.80 + 2.07i)3-s + (0.841 + 0.540i)4-s + (0.459 − 3.19i)5-s + (−2.31 + 1.48i)6-s + (−0.497 − 1.08i)7-s + (0.654 + 0.755i)8-s + (−0.649 − 4.51i)9-s + (1.34 − 2.93i)10-s + (−0.544 + 0.159i)11-s + (−2.63 + 0.774i)12-s + (−2.44 + 5.36i)13-s + (−0.170 − 1.18i)14-s + (5.81 + 6.70i)15-s + (0.415 + 0.909i)16-s + (−0.127 + 0.0817i)17-s + ⋯ |
| L(s) = 1 | + (0.678 + 0.199i)2-s + (−1.03 + 1.19i)3-s + (0.420 + 0.270i)4-s + (0.205 − 1.42i)5-s + (−0.944 + 0.606i)6-s + (−0.187 − 0.411i)7-s + (0.231 + 0.267i)8-s + (−0.216 − 1.50i)9-s + (0.424 − 0.928i)10-s + (−0.164 + 0.0481i)11-s + (−0.761 + 0.223i)12-s + (−0.678 + 1.48i)13-s + (−0.0455 − 0.316i)14-s + (1.50 + 1.73i)15-s + (0.103 + 0.227i)16-s + (−0.0308 + 0.0198i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 - 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.819504 + 0.288581i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.819504 + 0.288581i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.959 - 0.281i)T \) |
| 23 | \( 1 + (-4.67 + 1.07i)T \) |
| good | 3 | \( 1 + (1.80 - 2.07i)T + (-0.426 - 2.96i)T^{2} \) |
| 5 | \( 1 + (-0.459 + 3.19i)T + (-4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (0.497 + 1.08i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (0.544 - 0.159i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (2.44 - 5.36i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (0.127 - 0.0817i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (3.91 + 2.51i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-3.56 + 2.29i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-2.64 - 3.05i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-1.33 - 9.31i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.673 + 4.68i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (2.29 - 2.64i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 2.38T + 47T^{2} \) |
| 53 | \( 1 + (1.21 + 2.66i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (0.411 - 0.901i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (1.33 + 1.53i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-1.25 - 0.368i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (1.48 + 0.434i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (3.24 + 2.08i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-3.52 + 7.71i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (0.787 + 5.47i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (11.5 - 13.3i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-0.420 + 2.92i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−16.15144909929568226007871888934, −15.07527313983669916755760416249, −13.56681887315950982860301003993, −12.38131602474743131372408064158, −11.41428917421563954577948546907, −10.05176514053982912677684809728, −8.855427370940440549149268040529, −6.56113108777012281603866638524, −4.96567925681894845956541314273, −4.42509720885890363764982916985,
2.69664575449633268986665157642, 5.56430799547278332103524847504, 6.52118775130167478013110048034, 7.59313106412927177822036062180, 10.33776182870874853917255828454, 11.15284149932670494632728424809, 12.37087271003175599595409360478, 13.09138439039686867962926590938, 14.45279115810393141125480948027, 15.40340631555060190303990565413
