| L(s) = 1 | + (−0.173 + 0.984i)2-s + (−1.63 + 0.572i)3-s + (−0.939 − 0.342i)4-s + (0.789 − 0.662i)5-s + (−0.279 − 1.70i)6-s + (0.939 − 0.342i)7-s + (0.5 − 0.866i)8-s + (2.34 − 1.87i)9-s + (0.515 + 0.892i)10-s + (0.883 + 0.741i)11-s + (1.73 + 0.0212i)12-s + (1.14 + 6.46i)13-s + (0.173 + 0.984i)14-s + (−0.911 + 1.53i)15-s + (0.766 + 0.642i)16-s + (2.30 + 3.99i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 + 0.696i)2-s + (−0.943 + 0.330i)3-s + (−0.469 − 0.171i)4-s + (0.352 − 0.296i)5-s + (−0.114 − 0.697i)6-s + (0.355 − 0.129i)7-s + (0.176 − 0.306i)8-s + (0.781 − 0.623i)9-s + (0.162 + 0.282i)10-s + (0.266 + 0.223i)11-s + (0.499 + 0.00614i)12-s + (0.316 + 1.79i)13-s + (0.0464 + 0.263i)14-s + (−0.235 + 0.396i)15-s + (0.191 + 0.160i)16-s + (0.559 + 0.968i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0794 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0794 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.666384 + 0.721574i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.666384 + 0.721574i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 3 | \( 1 + (1.63 - 0.572i)T \) |
| 7 | \( 1 + (-0.939 + 0.342i)T \) |
| good | 5 | \( 1 + (-0.789 + 0.662i)T + (0.868 - 4.92i)T^{2} \) |
| 11 | \( 1 + (-0.883 - 0.741i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.14 - 6.46i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-2.30 - 3.99i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.93 + 3.34i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (8.49 + 3.09i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.0227 - 0.128i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-7.92 - 2.88i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-2.22 - 3.85i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.799 - 4.53i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-4.97 - 4.17i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-9.80 + 3.56i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 3.66T + 53T^{2} \) |
| 59 | \( 1 + (6.97 - 5.85i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-6.63 + 2.41i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (0.691 + 3.92i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (7.62 + 13.2i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.07 + 8.79i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.154 + 0.878i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (0.537 - 3.04i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-6.10 + 10.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.22 + 4.38i)T + (16.8 + 95.5i)T^{2} \) |
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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
−11.66811177084034784574792943912, −10.60028037205557259405036278931, −9.696748443246348777346033042083, −8.964614197109336216319864612079, −7.76027090730254416016916623226, −6.57736827409704024029348332853, −6.02232642749333291545608559497, −4.77253424882404090066168506625, −4.08445940263211436170165293121, −1.47873095029066946524144076529,
0.896171007083082602006389573331, 2.51133283093609623522940595740, 4.01599812501268635516833887010, 5.48375727473473522936252632071, 5.96393091486889884302344035270, 7.53369938696470448567090837428, 8.192026012922533118495042037274, 9.788697951527672246394845881418, 10.26588759252815231228838923304, 11.15419918328307259630134858793
