| L(s) = 1 | + (0.104 + 0.994i)2-s + (−0.0896 − 0.0995i)3-s + (−0.978 + 0.207i)4-s + (−1.17 + 1.89i)5-s + (0.0896 − 0.0995i)6-s + 2.27·7-s + (−0.309 − 0.951i)8-s + (0.311 − 2.96i)9-s + (−2.01 − 0.974i)10-s + (2.52 − 1.83i)11-s + (0.108 + 0.0787i)12-s + (0.459 − 4.36i)13-s + (0.238 + 2.26i)14-s + (0.294 − 0.0529i)15-s + (0.913 − 0.406i)16-s + (−1.75 − 0.373i)17-s + ⋯ |
| L(s) = 1 | + (0.0739 + 0.703i)2-s + (−0.0517 − 0.0574i)3-s + (−0.489 + 0.103i)4-s + (−0.527 + 0.849i)5-s + (0.0366 − 0.0406i)6-s + 0.860·7-s + (−0.109 − 0.336i)8-s + (0.103 − 0.988i)9-s + (−0.636 − 0.308i)10-s + (0.760 − 0.552i)11-s + (0.0312 + 0.0227i)12-s + (0.127 − 1.21i)13-s + (0.0636 + 0.605i)14-s + (0.0761 − 0.0136i)15-s + (0.228 − 0.101i)16-s + (−0.426 − 0.0907i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.52378 + 0.201072i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.52378 + 0.201072i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.104 - 0.994i)T \) |
| 5 | \( 1 + (1.17 - 1.89i)T \) |
| 19 | \( 1 + (2.13 + 3.80i)T \) |
| good | 3 | \( 1 + (0.0896 + 0.0995i)T + (-0.313 + 2.98i)T^{2} \) |
| 7 | \( 1 - 2.27T + 7T^{2} \) |
| 11 | \( 1 + (-2.52 + 1.83i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.459 + 4.36i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (1.75 + 0.373i)T + (15.5 + 6.91i)T^{2} \) |
| 23 | \( 1 + (-6.89 - 3.07i)T + (15.3 + 17.0i)T^{2} \) |
| 29 | \( 1 + (-2.31 + 0.492i)T + (26.4 - 11.7i)T^{2} \) |
| 31 | \( 1 + (-0.965 - 2.97i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (0.520 + 0.377i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-6.05 + 2.69i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (2.17 - 3.77i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.233 + 0.0495i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-6.99 + 1.48i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (2.10 - 0.936i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-6.06 - 2.70i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (-6.67 + 7.41i)T + (-7.00 - 66.6i)T^{2} \) |
| 71 | \( 1 + (-3.09 - 3.44i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (-1.39 - 13.2i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (5.94 + 6.59i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (3.32 + 10.2i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (2.17 + 0.970i)T + (59.5 + 66.1i)T^{2} \) |
| 97 | \( 1 + (-1.70 - 1.89i)T + (-10.1 + 96.4i)T^{2} \) |
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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
−10.04800908743096839282532419228, −8.943988839566815626098184384068, −8.394299474526744846010023787481, −7.36786138201251259680206381107, −6.77048169129416503108963882566, −5.93995456846087562304247445000, −4.85608019138324754175677269034, −3.80795520674943635346300258581, −2.91785577267221314757097337549, −0.844591838055836338245550193567,
1.32276451483195698227401134628, 2.21437622974948251485005070632, 3.99764096724588679281207408802, 4.52052000249675917190529692544, 5.22372571414605162481846705631, 6.64082276605754667213391548702, 7.70461727877985909604834379591, 8.532786964621948066241362814422, 9.087896392198784856416261195085, 10.07777715103418985898369224043
