| L(s) = 1 | + (0.309 + 0.951i)2-s + (0.978 − 0.207i)3-s + (−0.809 + 0.587i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s + (3.19 + 1.42i)7-s + (−0.809 − 0.587i)8-s + (0.913 − 0.406i)9-s + (0.978 + 0.207i)10-s + (0.412 + 3.92i)11-s + (−0.669 + 0.743i)12-s + (1.98 + 2.19i)13-s + (−0.365 + 3.47i)14-s + (0.309 − 0.951i)15-s + (0.309 − 0.951i)16-s + (0.838 − 7.97i)17-s + ⋯ |
| L(s) = 1 | + (0.218 + 0.672i)2-s + (0.564 − 0.120i)3-s + (−0.404 + 0.293i)4-s + (0.223 − 0.387i)5-s + (0.204 + 0.353i)6-s + (1.20 + 0.536i)7-s + (−0.286 − 0.207i)8-s + (0.304 − 0.135i)9-s + (0.309 + 0.0657i)10-s + (0.124 + 1.18i)11-s + (−0.193 + 0.214i)12-s + (0.549 + 0.610i)13-s + (−0.0975 + 0.928i)14-s + (0.0797 − 0.245i)15-s + (0.0772 − 0.237i)16-s + (0.203 − 1.93i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.463 - 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.463 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.09693 + 1.27021i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.09693 + 1.27021i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.309 - 0.951i)T \) |
| 3 | \( 1 + (-0.978 + 0.207i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (-2.58 + 4.92i)T \) |
| good | 7 | \( 1 + (-3.19 - 1.42i)T + (4.68 + 5.20i)T^{2} \) |
| 11 | \( 1 + (-0.412 - 3.92i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (-1.98 - 2.19i)T + (-1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.838 + 7.97i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (2.35 - 2.61i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (4.87 + 3.54i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.43 - 4.41i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-4.86 - 8.43i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-10.0 - 2.14i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (4.77 - 5.30i)T + (-4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (1.34 - 4.13i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-8.36 + 3.72i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (13.3 - 2.83i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + (-3.28 + 5.69i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-11.9 + 5.31i)T + (47.5 - 52.7i)T^{2} \) |
| 73 | \( 1 + (-0.00255 - 0.0243i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (0.200 - 1.90i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (10.3 + 2.19i)T + (75.8 + 33.7i)T^{2} \) |
| 89 | \( 1 + (-2.70 + 1.96i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (1.10 - 0.803i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.782469469777337606644440352960, −9.313946573954185599591918984587, −8.306461608857000555007524334019, −7.85929495003866798720332218512, −6.86731992404595133789058094074, −5.92686258915505233502436763825, −4.69137794604743336638242964895, −4.43762767814594786851045554956, −2.69365669583867004586900201670, −1.57372287452507913464702330902,
1.22560005735725640792319212510, 2.35050017443071070190374456842, 3.63905364093936273550462773721, 4.17543528704573675386130954669, 5.53455582798367523945405953050, 6.27238616754135105737562856639, 7.77141487309980428219463060226, 8.267590425446398499923971425819, 9.037520591946274688835148241264, 10.25655191708757346948301985129
