| L(s) = 1 | + (−0.809 − 0.587i)2-s + (−0.104 − 0.994i)3-s + (0.309 + 0.951i)4-s + (0.704 + 1.22i)5-s + (−0.5 + 0.866i)6-s + (3.13 + 0.667i)7-s + (0.309 − 0.951i)8-s + (−0.978 + 0.207i)9-s + (0.147 − 1.40i)10-s + (1.72 − 1.91i)11-s + (0.913 − 0.406i)12-s + (1.36 + 0.605i)13-s + (−2.14 − 2.38i)14-s + (1.13 − 0.828i)15-s + (−0.809 + 0.587i)16-s + (−4.31 − 4.78i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (−0.0603 − 0.574i)3-s + (0.154 + 0.475i)4-s + (0.315 + 0.545i)5-s + (−0.204 + 0.353i)6-s + (1.18 + 0.252i)7-s + (0.109 − 0.336i)8-s + (−0.326 + 0.0693i)9-s + (0.0465 − 0.443i)10-s + (0.519 − 0.576i)11-s + (0.263 − 0.117i)12-s + (0.377 + 0.167i)13-s + (−0.573 − 0.637i)14-s + (0.294 − 0.213i)15-s + (−0.202 + 0.146i)16-s + (−1.04 − 1.16i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.968451 - 0.352154i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.968451 - 0.352154i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 3 | \( 1 + (0.104 + 0.994i)T \) |
| 31 | \( 1 + (0.959 - 5.48i)T \) |
| good | 5 | \( 1 + (-0.704 - 1.22i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-3.13 - 0.667i)T + (6.39 + 2.84i)T^{2} \) |
| 11 | \( 1 + (-1.72 + 1.91i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-1.36 - 0.605i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (4.31 + 4.78i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-4.20 + 1.87i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (1.69 - 5.21i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-1.37 - 1.00i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (2.43 - 4.22i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.254 + 2.41i)T + (-40.1 - 8.52i)T^{2} \) |
| 43 | \( 1 + (6.61 - 2.94i)T + (28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (-1.16 + 0.846i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (8.87 - 1.88i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-1.09 - 10.4i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + 15.0T + 61T^{2} \) |
| 67 | \( 1 + (4.40 + 7.63i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-10.4 + 2.21i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (-5.25 + 5.83i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (2.91 + 3.24i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (0.475 - 4.52i)T + (-81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (-0.147 - 0.453i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (4.28 + 13.2i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−12.09689263462499823200486687738, −11.44766089480218636295377466367, −10.79256793677147934476979109637, −9.377037653617513241760629071927, −8.525937955757198237870428440248, −7.43953801478362782735557576060, −6.44084196369698897294979642046, −4.94896295171189342074344220465, −3.02638086916258215379506881510, −1.55151468771545015497360507485,
1.67794011070527521819780316875, 4.18051899795440260395942389775, 5.19469789075629091737283094855, 6.43225908773994241149092823234, 7.86896518241469457898710634265, 8.672764385666600466349550089269, 9.611335775367900909734119298270, 10.65374591729341783372855735112, 11.42878045117346265886785224058, 12.66553475514728200230105066336
