| L(s) = 1 | + (−0.541 + 1.30i)2-s + (−0.311 + 0.466i)3-s + (−1.41 − 1.41i)4-s + (−0.436 − 2.19i)5-s + (−0.440 − 0.659i)6-s + (−0.717 + 3.60i)7-s + (2.61 − 1.08i)8-s + (3.32 + 8.02i)9-s + (3.10 + 0.616i)10-s + (−10.6 + 7.12i)11-s + (1.09 − 0.218i)12-s + (−15.0 + 15.0i)13-s + (−4.32 − 2.88i)14-s + (1.15 + 0.479i)15-s + 4i·16-s + (−16.1 − 5.43i)17-s + ⋯ |
| L(s) = 1 | + (−0.270 + 0.653i)2-s + (−0.103 + 0.155i)3-s + (−0.353 − 0.353i)4-s + (−0.0872 − 0.438i)5-s + (−0.0734 − 0.109i)6-s + (−0.102 + 0.515i)7-s + (0.326 − 0.135i)8-s + (0.369 + 0.891i)9-s + (0.310 + 0.0616i)10-s + (−0.969 + 0.647i)11-s + (0.0916 − 0.0182i)12-s + (−1.16 + 1.16i)13-s + (−0.308 − 0.206i)14-s + (0.0771 + 0.0319i)15-s + 0.250i·16-s + (−0.947 − 0.319i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.349i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.124602 + 0.690168i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.124602 + 0.690168i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.541 - 1.30i)T \) |
| 5 | \( 1 + (0.436 + 2.19i)T \) |
| 17 | \( 1 + (16.1 + 5.43i)T \) |
| good | 3 | \( 1 + (0.311 - 0.466i)T + (-3.44 - 8.31i)T^{2} \) |
| 7 | \( 1 + (0.717 - 3.60i)T + (-45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (10.6 - 7.12i)T + (46.3 - 111. i)T^{2} \) |
| 13 | \( 1 + (15.0 - 15.0i)T - 169iT^{2} \) |
| 19 | \( 1 + (-8.38 + 20.2i)T + (-255. - 255. i)T^{2} \) |
| 23 | \( 1 + (-16.7 - 25.0i)T + (-202. + 488. i)T^{2} \) |
| 29 | \( 1 + (6.56 - 1.30i)T + (776. - 321. i)T^{2} \) |
| 31 | \( 1 + (3.39 + 2.26i)T + (367. + 887. i)T^{2} \) |
| 37 | \( 1 + (36.9 - 55.3i)T + (-523. - 1.26e3i)T^{2} \) |
| 41 | \( 1 + (-6.63 + 33.3i)T + (-1.55e3 - 643. i)T^{2} \) |
| 43 | \( 1 + (-29.7 - 71.8i)T + (-1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-47.7 + 47.7i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-14.6 + 35.4i)T + (-1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (61.8 - 25.6i)T + (2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-22.7 - 4.52i)T + (3.43e3 + 1.42e3i)T^{2} \) |
| 67 | \( 1 + 74.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-49.6 + 74.2i)T + (-1.92e3 - 4.65e3i)T^{2} \) |
| 73 | \( 1 + (-17.1 - 86.3i)T + (-4.92e3 + 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-111. + 74.4i)T + (2.38e3 - 5.76e3i)T^{2} \) |
| 83 | \( 1 + (-70.9 - 29.4i)T + (4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (38.4 + 38.4i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (115. - 23.0i)T + (8.69e3 - 3.60e3i)T^{2} \) |
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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
−13.15929287960627945393590737513, −12.01230301768819052106330075161, −10.89803569937619488591318759785, −9.673220383856482058047071525177, −9.013723330421380605200561525157, −7.65338826206585079391459396379, −6.93550589682080622079268127302, −5.18648812332348851769318882496, −4.69010302775259067928600941112, −2.24392600450998772939027715537,
0.45618473644393514520495908963, 2.64974244733519235261987120415, 3.90955302674782902733574893121, 5.53314902113788450362737208667, 7.02040483190952267472308492569, 7.941443271338005666937468012446, 9.223096905254489957908934813181, 10.43180057396103976145220100308, 10.76879542138616797862803192577, 12.26774072913469136443730811300
