L(s) = 1 | + (−0.268 + 1.38i)2-s + (−1.85 − 0.745i)4-s + (0.280 − 0.334i)5-s + (−1.12 + 0.646i)7-s + (1.53 − 2.37i)8-s + (0.389 + 0.479i)10-s + (0.842 − 1.45i)11-s + (0.944 + 5.35i)13-s + (−0.597 − 1.72i)14-s + (2.88 + 2.76i)16-s + (0.614 + 1.68i)17-s + (−3.19 + 2.96i)19-s + (−0.770 + 0.411i)20-s + (1.80 + 1.56i)22-s + (−1.92 + 1.61i)23-s + ⋯ |
L(s) = 1 | + (−0.189 + 0.981i)2-s + (−0.927 − 0.372i)4-s + (0.125 − 0.149i)5-s + (−0.423 + 0.244i)7-s + (0.541 − 0.840i)8-s + (0.123 + 0.151i)10-s + (0.254 − 0.439i)11-s + (0.261 + 1.48i)13-s + (−0.159 − 0.462i)14-s + (0.722 + 0.691i)16-s + (0.149 + 0.409i)17-s + (−0.733 + 0.680i)19-s + (−0.172 + 0.0920i)20-s + (0.383 + 0.332i)22-s + (−0.401 + 0.337i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.847 - 0.530i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.258888 + 0.901302i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.258888 + 0.901302i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.268 - 1.38i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (3.19 - 2.96i)T \) |
good | 5 | \( 1 + (-0.280 + 0.334i)T + (-0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (1.12 - 0.646i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.842 + 1.45i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.944 - 5.35i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.614 - 1.68i)T + (-13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (1.92 - 1.61i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.0255 + 0.0702i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (2.87 - 1.66i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 5.65T + 37T^{2} \) |
| 41 | \( 1 + (6.88 + 1.21i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (3.06 - 3.65i)T + (-7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-10.3 - 3.76i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (2.66 + 3.17i)T + (-9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (0.983 - 0.357i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.05 + 0.889i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (4.42 - 12.1i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-3.20 - 2.68i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-1.72 + 9.78i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-3.47 - 0.612i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (4.00 + 6.93i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.01 + 0.355i)T + (83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-4.95 + 1.80i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.65259783208783390560012641698, −9.631656855535684007684187588161, −9.039208250820206041804296914736, −8.296816735456854185283498432915, −7.24221237569561235842445182999, −6.35063627133078789716236428314, −5.75636989094623398691829697414, −4.50966830528751006492845161860, −3.58996575188174865480996265021, −1.60525017653250923387957942924,
0.55021876856309131387779260205, 2.26699144791559786447704615692, 3.28288281477525481891979077580, 4.33370574353556470339245064619, 5.40555251694558760291649594322, 6.59687337815481474769391941629, 7.73962836381501561430884098477, 8.551435924245651441194492061306, 9.491444847330310379556035118703, 10.29039714794948851241803623889