L(s) = 1 | + (1.61 + 1.17i)2-s + (−0.309 + 0.951i)3-s + (0.618 + 1.90i)4-s + (−3.23 + 2.35i)5-s + (−1.61 + 1.17i)6-s + (0.309 + 0.951i)7-s + (−0.809 − 0.587i)9-s − 8·10-s − 2.00·12-s + (1.61 + 1.17i)13-s + (−0.618 + 1.90i)14-s + (−1.23 − 3.80i)15-s + (3.23 − 2.35i)16-s + (−3.23 + 2.35i)17-s + (−0.618 − 1.90i)18-s + (−0.927 + 2.85i)19-s + ⋯ |
L(s) = 1 | + (1.14 + 0.831i)2-s + (−0.178 + 0.549i)3-s + (0.309 + 0.951i)4-s + (−1.44 + 1.05i)5-s + (−0.660 + 0.479i)6-s + (0.116 + 0.359i)7-s + (−0.269 − 0.195i)9-s − 2.52·10-s − 0.577·12-s + (0.448 + 0.326i)13-s + (−0.165 + 0.508i)14-s + (−0.319 − 0.982i)15-s + (0.809 − 0.587i)16-s + (−0.784 + 0.570i)17-s + (−0.145 − 0.448i)18-s + (−0.212 + 0.654i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.322131 + 1.65846i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.322131 + 1.65846i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-1.61 - 1.17i)T + (0.618 + 1.90i)T^{2} \) |
| 5 | \( 1 + (3.23 - 2.35i)T + (1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (-0.309 - 0.951i)T + (-5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (-1.61 - 1.17i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (3.23 - 2.35i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (0.927 - 2.85i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 + (-1.85 - 5.70i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-4.04 - 2.93i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.927 - 2.85i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (0.618 - 1.90i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 12T + 43T^{2} \) |
| 47 | \( 1 + (-0.618 + 1.90i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (4.85 + 3.52i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (3.09 + 9.51i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (2.42 - 1.76i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + T + 67T^{2} \) |
| 71 | \( 1 + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (3.39 + 10.4i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (8.89 + 6.46i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (4.85 - 3.52i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + (4.04 + 2.93i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.92353942947076934449411595522, −11.04885348134601243696081674513, −10.33393897882517166315596108053, −8.772710341791406750767467751219, −7.79657745460773506535140235869, −6.81181152174878746713040403945, −6.12776906322719890434001836234, −4.77743798696406079333418042085, −3.97961642880742711257925137030, −3.12504292312340698071910735657,
0.872597404376471983115521268714, 2.72506616924827748787200966047, 4.14403782812745651104495231310, 4.59065229239504166699727938150, 5.81396678845320349125158428825, 7.30393750512752986916598454666, 8.128884021528328490221600093320, 9.047948435097816093092014505107, 10.76764768018248699985338455082, 11.37896408886261441260111258555