L(s) = 1 | + (0.154 + 0.0744i)2-s + (−1.22 − 1.54i)4-s − 0.566·5-s + (−1.38 + 0.668i)7-s + (−0.151 − 0.664i)8-s + (−0.0876 − 0.0421i)10-s + (−0.879 + 3.85i)11-s + (−0.834 − 3.65i)13-s − 0.264·14-s + (−0.850 + 3.72i)16-s + 4.10·17-s + (−2.81 + 3.53i)19-s + (0.696 + 0.873i)20-s + (−0.422 + 0.530i)22-s + (−7.97 + 3.83i)23-s + ⋯ |
L(s) = 1 | + (0.109 + 0.0526i)2-s + (−0.614 − 0.770i)4-s − 0.253·5-s + (−0.524 + 0.252i)7-s + (−0.0536 − 0.234i)8-s + (−0.0277 − 0.0133i)10-s + (−0.265 + 1.16i)11-s + (−0.231 − 1.01i)13-s − 0.0706·14-s + (−0.212 + 0.932i)16-s + 0.994·17-s + (−0.646 + 0.810i)19-s + (0.155 + 0.195i)20-s + (−0.0901 + 0.113i)22-s + (−1.66 + 0.800i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.649 - 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.649 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.149075 + 0.323556i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.149075 + 0.323556i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 71 | \( 1 + (-1.69 - 8.25i)T \) |
good | 2 | \( 1 + (-0.154 - 0.0744i)T + (1.24 + 1.56i)T^{2} \) |
| 5 | \( 1 + 0.566T + 5T^{2} \) |
| 7 | \( 1 + (1.38 - 0.668i)T + (4.36 - 5.47i)T^{2} \) |
| 11 | \( 1 + (0.879 - 3.85i)T + (-9.91 - 4.77i)T^{2} \) |
| 13 | \( 1 + (0.834 + 3.65i)T + (-11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 - 4.10T + 17T^{2} \) |
| 19 | \( 1 + (2.81 - 3.53i)T + (-4.22 - 18.5i)T^{2} \) |
| 23 | \( 1 + (7.97 - 3.83i)T + (14.3 - 17.9i)T^{2} \) |
| 29 | \( 1 + (-0.0365 - 0.0458i)T + (-6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (-2.08 - 9.12i)T + (-27.9 + 13.4i)T^{2} \) |
| 37 | \( 1 + (-0.457 - 0.220i)T + (23.0 + 28.9i)T^{2} \) |
| 41 | \( 1 + (1.72 + 7.56i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (-3.32 - 1.59i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (3.55 - 4.46i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (0.685 - 0.859i)T + (-11.7 - 51.6i)T^{2} \) |
| 59 | \( 1 + (-0.774 + 3.39i)T + (-53.1 - 25.5i)T^{2} \) |
| 61 | \( 1 + (-1.38 - 0.668i)T + (38.0 + 47.6i)T^{2} \) |
| 67 | \( 1 + (2.63 + 3.30i)T + (-14.9 + 65.3i)T^{2} \) |
| 73 | \( 1 + (11.4 + 5.49i)T + (45.5 + 57.0i)T^{2} \) |
| 79 | \( 1 + (3.47 + 15.2i)T + (-71.1 + 34.2i)T^{2} \) |
| 83 | \( 1 + (-2.87 + 12.5i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (-0.321 + 0.403i)T + (-19.8 - 86.7i)T^{2} \) |
| 97 | \( 1 + (-1.43 + 6.28i)T + (-87.3 - 42.0i)T^{2} \) |
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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
−10.43120814832495951270808421376, −10.12375427784725844242517492880, −9.388160560931638790518407127853, −8.211703101130008558592555255958, −7.47749950412178145075879619102, −6.15276666072366611790994760924, −5.51409870817271103158062422106, −4.46257559475134761505104434131, −3.39132970430088303091278461368, −1.74999036577992164517833798935,
0.18379103936424101108744028408, 2.54309456463266207355859842705, 3.72503211382564591917563609067, 4.37525669381874417037110697824, 5.72532232672053128769832628513, 6.69017538251401520354271565869, 7.87906479107893986558392642598, 8.355212375693004059555259909784, 9.414422533506274563240067284488, 10.09958853014304276768447477782