| L(s) = 1 | + (0.977 + 0.212i)2-s + (0.737 + 0.985i)3-s + (0.909 + 0.415i)4-s + (−0.103 + 2.23i)5-s + (0.511 + 1.11i)6-s + (−1.20 − 0.0864i)7-s + (0.800 + 0.599i)8-s + (0.418 − 1.42i)9-s + (−0.575 + 2.16i)10-s + (−1.71 + 2.66i)11-s + (0.261 + 1.20i)12-s + (−0.366 − 5.12i)13-s + (−1.16 − 0.341i)14-s + (−2.27 + 1.54i)15-s + (0.654 + 0.755i)16-s + (3.41 + 1.27i)17-s + ⋯ |
| L(s) = 1 | + (0.690 + 0.150i)2-s + (0.426 + 0.569i)3-s + (0.454 + 0.207i)4-s + (−0.0461 + 0.998i)5-s + (0.208 + 0.457i)6-s + (−0.456 − 0.0326i)7-s + (0.283 + 0.211i)8-s + (0.139 − 0.474i)9-s + (−0.182 + 0.683i)10-s + (−0.516 + 0.804i)11-s + (0.0755 + 0.347i)12-s + (−0.101 − 1.42i)13-s + (−0.310 − 0.0912i)14-s + (−0.588 + 0.399i)15-s + (0.163 + 0.188i)16-s + (0.827 + 0.308i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.473 - 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.473 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.65958 + 0.992583i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.65958 + 0.992583i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.977 - 0.212i)T \) |
| 5 | \( 1 + (0.103 - 2.23i)T \) |
| 23 | \( 1 + (-3.12 + 3.64i)T \) |
| good | 3 | \( 1 + (-0.737 - 0.985i)T + (-0.845 + 2.87i)T^{2} \) |
| 7 | \( 1 + (1.20 + 0.0864i)T + (6.92 + 0.996i)T^{2} \) |
| 11 | \( 1 + (1.71 - 2.66i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (0.366 + 5.12i)T + (-12.8 + 1.85i)T^{2} \) |
| 17 | \( 1 + (-3.41 - 1.27i)T + (12.8 + 11.1i)T^{2} \) |
| 19 | \( 1 + (-0.552 + 1.21i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (0.381 - 0.174i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (0.00687 + 0.0478i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-6.11 + 3.33i)T + (20.0 - 31.1i)T^{2} \) |
| 41 | \( 1 + (7.66 - 2.25i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (9.47 - 7.09i)T + (12.1 - 41.2i)T^{2} \) |
| 47 | \( 1 + (0.789 + 0.789i)T + 47iT^{2} \) |
| 53 | \( 1 + (-0.739 + 10.3i)T + (-52.4 - 7.54i)T^{2} \) |
| 59 | \( 1 + (-1.43 - 1.24i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-3.22 + 0.463i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (1.04 - 4.80i)T + (-60.9 - 27.8i)T^{2} \) |
| 71 | \( 1 + (6.59 - 4.23i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (4.16 + 11.1i)T + (-55.1 + 47.8i)T^{2} \) |
| 79 | \( 1 + (9.78 - 11.2i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-2.97 - 5.45i)T + (-44.8 + 69.8i)T^{2} \) |
| 89 | \( 1 + (0.260 - 1.81i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-1.28 + 2.35i)T + (-52.4 - 81.6i)T^{2} \) |
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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
−12.61931354350041172906306285585, −11.43332467397147598290534953753, −10.20499777975567925236818079567, −9.918067019035872393774949716222, −8.262653237429063910324526276965, −7.22297942786680354807842199176, −6.25023581343858790956559301140, −4.95542945000636404834035017258, −3.53786601778821614913111053082, −2.81391212378235485096977295100,
1.63052941706895328914145599736, 3.21315914074305513532302686597, 4.66006945645858531806329353192, 5.67701680364109534011180660046, 6.99727767502383905870545016677, 8.016419773401813151494775273781, 9.026601355685524255741526838146, 10.10934477659677037232558452995, 11.46915601557256191930432226930, 12.16797812683873596997510306388
