| L(s) = 1 | + (−1.32 + 1.53i)2-s + (0.841 − 0.540i)3-s + (−0.300 − 2.08i)4-s + (−0.415 − 0.909i)5-s + (−0.288 + 2.00i)6-s + (3.59 − 1.05i)7-s + (0.189 + 0.121i)8-s + (0.415 − 0.909i)9-s + (1.94 + 0.571i)10-s + (−3.62 − 4.18i)11-s + (−1.38 − 1.59i)12-s + (−3.83 − 1.12i)13-s + (−3.15 + 6.91i)14-s + (−0.841 − 0.540i)15-s + (3.61 − 1.06i)16-s + (0.723 − 5.03i)17-s + ⋯ |
| L(s) = 1 | + (−0.938 + 1.08i)2-s + (0.485 − 0.312i)3-s + (−0.150 − 1.04i)4-s + (−0.185 − 0.406i)5-s + (−0.117 + 0.819i)6-s + (1.35 − 0.399i)7-s + (0.0670 + 0.0431i)8-s + (0.138 − 0.303i)9-s + (0.615 + 0.180i)10-s + (−1.09 − 1.26i)11-s + (−0.399 − 0.460i)12-s + (−1.06 − 0.312i)13-s + (−0.843 + 1.84i)14-s + (−0.217 − 0.139i)15-s + (0.903 − 0.265i)16-s + (0.175 − 1.22i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.867829 - 0.158815i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.867829 - 0.158815i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-3.81 + 2.89i)T \) |
| good | 2 | \( 1 + (1.32 - 1.53i)T + (-0.284 - 1.97i)T^{2} \) |
| 7 | \( 1 + (-3.59 + 1.05i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (3.62 + 4.18i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (3.83 + 1.12i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.723 + 5.03i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.272 - 1.89i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (0.784 - 5.45i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-0.152 - 0.0977i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-1.85 + 4.07i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-4.83 - 10.5i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-4.33 + 2.78i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 3.81T + 47T^{2} \) |
| 53 | \( 1 + (2.43 - 0.714i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-13.6 - 4.01i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (5.48 + 3.52i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (4.17 - 4.81i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (7.80 - 9.01i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (1.96 + 13.6i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (4.96 + 1.45i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-3.46 + 7.58i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (1.79 - 1.15i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-7.42 - 16.2i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−11.27936765071921198680021269526, −10.32152261427605421771469271903, −9.158321217086070572621288981878, −8.418326217958631172010706968298, −7.69739746188587094569007674302, −7.26353729495749560410825792479, −5.68997181379459362849487812696, −4.80914774435027775897985440702, −2.89182513090367755618804017992, −0.809088951041004113379280387708,
1.91623692641500957643000717308, 2.63565049140425518652368442493, 4.29937146466282258299236454323, 5.39411724982448268073132740139, 7.42024829240604235355561964005, 7.983554943888436485110182015305, 8.970057520315882888191775031733, 9.878983157026373153271865834488, 10.54298233558208729479029209329, 11.32532001162612368071142443188
