L(s) = 1 | + (0.241 + 0.970i)2-s + (0.0916 − 0.187i)3-s + (−0.882 + 0.469i)4-s + (0.913 − 0.406i)5-s + (0.204 + 0.0434i)6-s + (−0.939 − 1.62i)7-s + (−0.669 − 0.743i)8-s + (0.588 + 0.753i)9-s + (0.615 + 0.788i)10-s + (0.00729 + 0.208i)12-s + (1.35 − 1.30i)14-s + (0.00729 − 0.208i)15-s + (0.559 − 0.829i)16-s + (−0.588 + 0.753i)18-s + (−0.615 + 0.788i)20-s + (−0.391 + 0.0274i)21-s + ⋯ |
L(s) = 1 | + (0.241 + 0.970i)2-s + (0.0916 − 0.187i)3-s + (−0.882 + 0.469i)4-s + (0.913 − 0.406i)5-s + (0.204 + 0.0434i)6-s + (−0.939 − 1.62i)7-s + (−0.669 − 0.743i)8-s + (0.588 + 0.753i)9-s + (0.615 + 0.788i)10-s + (0.00729 + 0.208i)12-s + (1.35 − 1.30i)14-s + (0.00729 − 0.208i)15-s + (0.559 − 0.829i)16-s + (−0.588 + 0.753i)18-s + (−0.615 + 0.788i)20-s + (−0.391 + 0.0274i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.404696162\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.404696162\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.241 - 0.970i)T \) |
| 5 | \( 1 + (-0.913 + 0.406i)T \) |
| 181 | \( 1 + (0.882 - 0.469i)T \) |
good | 3 | \( 1 + (-0.0916 + 0.187i)T + (-0.615 - 0.788i)T^{2} \) |
| 7 | \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.438 - 0.898i)T^{2} \) |
| 13 | \( 1 + (-0.0348 - 0.999i)T^{2} \) |
| 17 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-0.328 - 0.812i)T + (-0.719 + 0.694i)T^{2} \) |
| 29 | \( 1 + (1.33 + 1.48i)T + (-0.104 + 0.994i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.990 + 0.139i)T^{2} \) |
| 41 | \( 1 + (-1.51 - 0.213i)T + (0.961 + 0.275i)T^{2} \) |
| 43 | \( 1 + (0.333 + 1.89i)T + (-0.939 + 0.342i)T^{2} \) |
| 47 | \( 1 + (0.0467 + 1.33i)T + (-0.997 + 0.0697i)T^{2} \) |
| 53 | \( 1 + (-0.961 + 0.275i)T^{2} \) |
| 59 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (-1.51 + 1.27i)T + (0.173 - 0.984i)T^{2} \) |
| 67 | \( 1 + (-1.78 - 0.379i)T + (0.913 + 0.406i)T^{2} \) |
| 71 | \( 1 + (0.978 - 0.207i)T^{2} \) |
| 73 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 79 | \( 1 + (-0.961 + 0.275i)T^{2} \) |
| 83 | \( 1 + (0.594 + 0.170i)T + (0.848 + 0.529i)T^{2} \) |
| 89 | \( 1 + (1.35 + 1.13i)T + (0.173 + 0.984i)T^{2} \) |
| 97 | \( 1 + (-0.438 + 0.898i)T^{2} \) |
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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
−8.567794357204104931228172467641, −7.69742705579854761416431309255, −7.19042346445716640881808066150, −6.62489756948638298719950280601, −5.77113891218550676512444141671, −5.07240785226806622648452244836, −4.13232637488911848024089390187, −3.60521666296040553981642966656, −2.19029414507194780538078788873, −0.796272406823512472722899510459,
1.41831744440605824089144794619, 2.52360650851270657096482031713, 2.96496618499959719124294818757, 3.86432813729848408617471167154, 4.97753166577419377506829705558, 5.72604113843547296036412573858, 6.24583018931208733467093630199, 7.01247013825156377051622610964, 8.458224195886942746284633351944, 9.092162449748914845349878179466