| L(s) = 1 | + (−0.309 + 0.951i)2-s + (0.669 − 0.743i)3-s + (−0.809 − 0.587i)4-s + (−0.5 + 0.866i)5-s + (0.499 + 0.866i)6-s + (−0.0662 + 0.630i)7-s + (0.809 − 0.587i)8-s + (−0.104 − 0.994i)9-s + (−0.669 − 0.743i)10-s + (4.10 − 1.82i)11-s + (−0.978 + 0.207i)12-s + (−0.188 − 0.0401i)13-s + (−0.579 − 0.257i)14-s + (0.309 + 0.951i)15-s + (0.309 + 0.951i)16-s + (−3.34 − 1.49i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + (0.386 − 0.429i)3-s + (−0.404 − 0.293i)4-s + (−0.223 + 0.387i)5-s + (0.204 + 0.353i)6-s + (−0.0250 + 0.238i)7-s + (0.286 − 0.207i)8-s + (−0.0348 − 0.331i)9-s + (−0.211 − 0.235i)10-s + (1.23 − 0.551i)11-s + (−0.282 + 0.0600i)12-s + (−0.0523 − 0.0111i)13-s + (−0.154 − 0.0689i)14-s + (0.0797 + 0.245i)15-s + (0.0772 + 0.237i)16-s + (−0.812 − 0.361i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.321i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.56084 + 0.257498i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.56084 + 0.257498i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.669 + 0.743i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (-4.61 - 3.11i)T \) |
| good | 7 | \( 1 + (0.0662 - 0.630i)T + (-6.84 - 1.45i)T^{2} \) |
| 11 | \( 1 + (-4.10 + 1.82i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (0.188 + 0.0401i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (3.34 + 1.49i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (-4.11 + 0.873i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-3.42 + 2.48i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (0.643 - 1.97i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (-1.86 - 3.22i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.000998 - 0.00110i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-6.41 + 1.36i)T + (39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (-0.500 - 1.54i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.161 + 1.53i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (-1.32 + 1.47i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 - 9.00T + 61T^{2} \) |
| 67 | \( 1 + (-1.19 + 2.07i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.115 - 1.09i)T + (-69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (-10.0 + 4.45i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (3.81 + 1.69i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (4.18 + 4.64i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (6.23 + 4.53i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (4.10 + 2.98i)T + (29.9 + 92.2i)T^{2} \) |
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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
−9.833634995540671371792888011516, −9.003505280858981434238708147212, −8.533064357319138640233860292909, −7.44389805783481329977876465933, −6.79366210144699220904420082069, −6.11945193775223170096974911582, −4.92182952267448774817580937114, −3.77603028878223909075159119310, −2.66672017636104544404219765578, −1.01895255368290405248482382395,
1.15839211941855911541231710396, 2.48729434900947735039691823113, 3.81913641020974736332658215991, 4.29083087487903587844950632831, 5.43877579583157871995809184213, 6.77505354868611927058550707976, 7.68248554795163229605462321402, 8.617192435906769145106247006730, 9.363123151742110896040521301481, 9.808344423212201823985208703504
