L(s) = 1 | + (−1.20 − 0.742i)2-s + (0.896 + 1.78i)4-s + (0.206 + 0.769i)5-s + (−2.17 − 3.76i)7-s + (0.248 − 2.81i)8-s + (0.323 − 1.07i)10-s + (−1.05 + 3.93i)11-s + (−0.454 − 1.69i)13-s + (−0.180 + 6.15i)14-s + (−2.39 + 3.20i)16-s − 6.68i·17-s + (−0.708 − 0.708i)19-s + (−1.19 + 1.05i)20-s + (4.19 − 3.95i)22-s + (−3.88 − 2.24i)23-s + ⋯ |
L(s) = 1 | + (−0.850 − 0.525i)2-s + (0.448 + 0.893i)4-s + (0.0922 + 0.344i)5-s + (−0.822 − 1.42i)7-s + (0.0879 − 0.996i)8-s + (0.102 − 0.341i)10-s + (−0.318 + 1.18i)11-s + (−0.126 − 0.470i)13-s + (−0.0482 + 1.64i)14-s + (−0.598 + 0.801i)16-s − 1.62i·17-s + (−0.162 − 0.162i)19-s + (−0.266 + 0.236i)20-s + (0.894 − 0.843i)22-s + (−0.809 − 0.467i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 + 0.629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.776 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.176358 - 0.497716i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.176358 - 0.497716i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.20 + 0.742i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.206 - 0.769i)T + (-4.33 + 2.5i)T^{2} \) |
| 7 | \( 1 + (2.17 + 3.76i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.05 - 3.93i)T + (-9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (0.454 + 1.69i)T + (-11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + 6.68iT - 17T^{2} \) |
| 19 | \( 1 + (0.708 + 0.708i)T + 19iT^{2} \) |
| 23 | \( 1 + (3.88 + 2.24i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.06 + 3.98i)T + (-25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (4.94 + 2.85i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.51 + 1.51i)T + 37iT^{2} \) |
| 41 | \( 1 + (1.36 - 2.36i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (8.60 + 2.30i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (1.23 + 2.13i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.68 - 1.68i)T - 53iT^{2} \) |
| 59 | \( 1 + (-1.00 + 0.269i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-1.97 - 0.528i)T + (52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (-8.01 + 2.14i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 8.05iT - 71T^{2} \) |
| 73 | \( 1 - 9.73iT - 73T^{2} \) |
| 79 | \( 1 + (-11.9 + 6.91i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.05 + 0.817i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + 1.71T + 89T^{2} \) |
| 97 | \( 1 + (-3.66 - 6.34i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.45933748806970462561011923052, −10.04702729734473223825708012552, −9.343256639744154012213571573761, −7.943171427200232872628037401793, −7.18566470712322005920684078451, −6.60731274655838007567689890078, −4.68986621478318231991006209739, −3.52822797688984691354683037213, −2.36379038811813220089046457402, −0.42169835079581065321551405893,
1.82385625363389864934436167458, 3.30595883487530145232127076236, 5.27703849404204516277878167442, 5.95626821953726590726296133138, 6.73939520002899022208080712026, 8.254262680493123447301631291464, 8.671732497231104375433329917061, 9.467657798119110033686256840421, 10.42494359323318612267625549913, 11.33323294173172703588487520247