| L(s) = 1 | + (0.309 + 0.951i)2-s + (0.669 + 0.743i)3-s + (−0.809 + 0.587i)4-s + (0.360 + 0.623i)5-s + (−0.499 + 0.866i)6-s + (0.173 + 1.65i)7-s + (−0.809 − 0.587i)8-s + (−0.104 + 0.994i)9-s + (−0.481 + 0.535i)10-s + (−1.28 − 0.573i)11-s + (−0.978 − 0.207i)12-s + (2.72 − 0.578i)13-s + (−1.51 + 0.675i)14-s + (−0.222 + 0.684i)15-s + (0.309 − 0.951i)16-s + (−0.388 + 0.173i)17-s + ⋯ |
| L(s) = 1 | + (0.218 + 0.672i)2-s + (0.386 + 0.429i)3-s + (−0.404 + 0.293i)4-s + (0.161 + 0.278i)5-s + (−0.204 + 0.353i)6-s + (0.0656 + 0.624i)7-s + (−0.286 − 0.207i)8-s + (−0.0348 + 0.331i)9-s + (−0.152 + 0.169i)10-s + (−0.388 − 0.172i)11-s + (−0.282 − 0.0600i)12-s + (0.755 − 0.160i)13-s + (−0.405 + 0.180i)14-s + (−0.0574 + 0.176i)15-s + (0.0772 − 0.237i)16-s + (−0.0943 + 0.0419i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0890 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0890 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.956036 + 1.04534i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.956036 + 1.04534i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.669 - 0.743i)T \) |
| 31 | \( 1 + (-4.91 + 2.60i)T \) |
| good | 5 | \( 1 + (-0.360 - 0.623i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.173 - 1.65i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (1.28 + 0.573i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (-2.72 + 0.578i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (0.388 - 0.173i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-2.25 - 0.479i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (4.74 + 3.44i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.0752 - 0.231i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-1.38 + 2.39i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.38 + 3.75i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (2.77 + 0.590i)T + (39.2 + 17.4i)T^{2} \) |
| 47 | \( 1 + (2.12 - 6.55i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.501 + 4.77i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (8.08 + 8.97i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + 2.61T + 61T^{2} \) |
| 67 | \( 1 + (-2.62 - 4.54i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.12 - 10.7i)T + (-69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (5.42 + 2.41i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (12.2 - 5.44i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (4.01 - 4.45i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (0.481 - 0.350i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (4.23 - 3.07i)T + (29.9 - 92.2i)T^{2} \) |
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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
−13.06982046530156333517667565923, −12.00092657835850957699840618274, −10.77192033860049549609665217699, −9.757297419907378144770370814762, −8.643816874031868670842749292419, −7.923921862826087328906340626165, −6.46375758054918651309351678480, −5.50648933310426402298917393688, −4.16734279726181498225964566195, −2.71414204795541806599038875188,
1.44040746368864567345129669594, 3.14665318292672388307425107665, 4.46774428079240490864101375618, 5.88120457263205919671430107209, 7.27301460973971476749326942280, 8.372902374778844706672245261578, 9.445175099840677565367424292647, 10.41829037492749924167037676638, 11.45987067123289687623454396268, 12.37995043058746134484690290644
