| L(s) = 1 | + (−0.342 − 0.939i)2-s + (1.51 + 0.847i)3-s + (−0.766 + 0.642i)4-s + (−0.0403 − 0.228i)5-s + (0.279 − 1.70i)6-s + (1.82 − 1.91i)7-s + (0.866 + 0.500i)8-s + (1.56 + 2.55i)9-s + (−0.201 + 0.116i)10-s + (2.79 + 0.492i)11-s + (−1.70 + 0.322i)12-s + (−1.29 + 3.56i)13-s + (−2.42 − 1.06i)14-s + (0.132 − 0.379i)15-s + (0.173 − 0.984i)16-s + (−3.85 − 6.67i)17-s + ⋯ |
| L(s) = 1 | + (−0.241 − 0.664i)2-s + (0.872 + 0.489i)3-s + (−0.383 + 0.321i)4-s + (−0.0180 − 0.102i)5-s + (0.114 − 0.697i)6-s + (0.690 − 0.723i)7-s + (0.306 + 0.176i)8-s + (0.521 + 0.853i)9-s + (−0.0636 + 0.0367i)10-s + (0.841 + 0.148i)11-s + (−0.491 + 0.0929i)12-s + (−0.359 + 0.987i)13-s + (−0.647 − 0.283i)14-s + (0.0342 − 0.0980i)15-s + (0.0434 − 0.246i)16-s + (−0.935 − 1.62i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.63280 - 0.359494i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.63280 - 0.359494i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.342 + 0.939i)T \) |
| 3 | \( 1 + (-1.51 - 0.847i)T \) |
| 7 | \( 1 + (-1.82 + 1.91i)T \) |
| good | 5 | \( 1 + (0.0403 + 0.228i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (-2.79 - 0.492i)T + (10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (1.29 - 3.56i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (3.85 + 6.67i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.59 - 1.49i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.84 - 3.38i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (2.66 + 7.31i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-0.0308 - 0.0367i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-5.34 - 9.25i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.22 + 1.90i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.0109 - 0.0621i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (7.18 + 6.02i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 10.8iT - 53T^{2} \) |
| 59 | \( 1 + (1.35 + 7.65i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (5.20 - 6.20i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (13.8 + 5.02i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (6.22 - 3.59i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (7.95 + 4.59i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (9.36 - 3.40i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (3.51 - 1.27i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (2.75 - 4.76i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.53 - 1.68i)T + (91.1 + 33.1i)T^{2} \) |
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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
−11.43571384867623627945615138966, −10.23230145400498204446216429226, −9.437515691177376624835931880539, −8.859330333697841430929646768333, −7.70751843322327597583969566880, −6.92566031262644403748282213137, −4.80963515851761506077918012964, −4.27459841924713271031577165033, −2.95277981230077951627846455471, −1.56980422985595043121037958902,
1.56588819047187209540073691440, 3.10276681451199369800203648636, 4.55778986536108085852092749261, 5.85592847762170377640153090112, 6.84908494646690683765745599392, 7.76328543865458478377088616106, 8.737645059556231258544261497641, 9.022262047086704063361489311715, 10.38914834557517532651083709911, 11.39095689700699069316433198220
