L(s) = 1 | + (−0.642 + 1.60i)3-s + (1.28 − 2.23i)5-s + (0.203 − 2.63i)7-s + (−2.17 − 2.06i)9-s + (1.43 − 0.826i)11-s − 5.71i·13-s + (2.76 + 3.50i)15-s + (3.79 + 6.56i)17-s + (−2.58 − 1.49i)19-s + (4.11 + 2.02i)21-s + (0.249 + 0.143i)23-s + (−0.825 − 1.43i)25-s + (4.72 − 2.16i)27-s + 2.05i·29-s + (5.21 − 3.00i)31-s + ⋯ |
L(s) = 1 | + (−0.371 + 0.928i)3-s + (0.576 − 0.998i)5-s + (0.0768 − 0.997i)7-s + (−0.724 − 0.689i)9-s + (0.431 − 0.249i)11-s − 1.58i·13-s + (0.713 + 0.906i)15-s + (0.919 + 1.59i)17-s + (−0.594 − 0.343i)19-s + (0.897 + 0.441i)21-s + (0.0519 + 0.0300i)23-s + (−0.165 − 0.286i)25-s + (0.908 − 0.417i)27-s + 0.382i·29-s + (0.936 − 0.540i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.834 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.20566 - 0.362307i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20566 - 0.362307i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.642 - 1.60i)T \) |
| 7 | \( 1 + (-0.203 + 2.63i)T \) |
good | 5 | \( 1 + (-1.28 + 2.23i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.43 + 0.826i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 5.71iT - 13T^{2} \) |
| 17 | \( 1 + (-3.79 - 6.56i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.58 + 1.49i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.249 - 0.143i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2.05iT - 29T^{2} \) |
| 31 | \( 1 + (-5.21 + 3.00i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.877 - 1.51i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 4.28T + 41T^{2} \) |
| 43 | \( 1 + 2.46T + 43T^{2} \) |
| 47 | \( 1 + (-0.186 + 0.323i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.73 - 3.88i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.89 + 8.48i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.889 - 0.513i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.18 - 2.04i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 15.6iT - 71T^{2} \) |
| 73 | \( 1 + (3.30 - 1.90i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.56 - 7.89i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 6.65T + 83T^{2} \) |
| 89 | \( 1 + (-7.25 + 12.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 4.43iT - 97T^{2} \) |
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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
−11.22811227788553747382750907880, −10.38000141971606750985034350488, −9.871599355052059887123990961276, −8.718999703460147405953944950956, −7.947451140087494299677852387073, −6.28197980877308278748843154787, −5.46128568045865471165382811365, −4.45292486738036702010474447560, −3.39679287658090198475772577259, −1.02123321135000902402846673603,
1.87435532066942530968591045522, 2.86845563748581082415577490268, 4.86242368697793892696606577149, 6.11682495291297444976858960661, 6.65422193562869872585556181848, 7.61365351196521109246471704978, 8.898115150053610575015434221755, 9.740004347832950164245090735814, 10.93109034766312146794818989059, 11.86962544232712801783867868299