| 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
−12.16797812683873596997510306388, −11.46915601557256191930432226930, −10.10934477659677037232558452995, −9.026601355685524255741526838146, −8.016419773401813151494775273781, −6.99727767502383905870545016677, −5.67701680364109534011180660046, −4.66006945645858531806329353192, −3.21315914074305513532302686597, −1.63052941706895328914145599736,
2.81391212378235485096977295100, 3.53786601778821614913111053082, 4.95542945000636404834035017258, 6.25023581343858790956559301140, 7.22297942786680354807842199176, 8.262653237429063910324526276965, 9.918067019035872393774949716222, 10.20499777975567925236818079567, 11.43332467397147598290534953753, 12.61931354350041172906306285585
