L(s) = 1 | + (0.655 + 1.25i)2-s + (−3.05 + 0.817i)3-s + (−1.14 + 1.64i)4-s + (0.213 − 0.797i)5-s + (−3.02 − 3.28i)6-s + (−2.80 − 0.353i)8-s + (6.05 − 3.49i)9-s + (1.13 − 0.254i)10-s + (2.73 − 0.732i)11-s + (2.13 − 5.94i)12-s + (2.91 − 2.91i)13-s + 2.61i·15-s + (−1.39 − 3.74i)16-s + (−2.30 − 1.33i)17-s + (8.34 + 5.29i)18-s + (0.117 − 0.436i)19-s + ⋯ |
L(s) = 1 | + (0.463 + 0.886i)2-s + (−1.76 + 0.472i)3-s + (−0.570 + 0.821i)4-s + (0.0956 − 0.356i)5-s + (−1.23 − 1.34i)6-s + (−0.992 − 0.125i)8-s + (2.01 − 1.16i)9-s + (0.360 − 0.0806i)10-s + (0.823 − 0.220i)11-s + (0.617 − 1.71i)12-s + (0.807 − 0.807i)13-s + 0.673i·15-s + (−0.348 − 0.937i)16-s + (−0.558 − 0.322i)17-s + (1.96 + 1.24i)18-s + (0.0268 − 0.100i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.853373 + 0.589650i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.853373 + 0.589650i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.655 - 1.25i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (3.05 - 0.817i)T + (2.59 - 1.5i)T^{2} \) |
| 5 | \( 1 + (-0.213 + 0.797i)T + (-4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (-2.73 + 0.732i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-2.91 + 2.91i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.30 + 1.33i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.117 + 0.436i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-1.63 - 2.83i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.04 + 2.04i)T + 29iT^{2} \) |
| 31 | \( 1 + (1.26 - 2.18i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.33 + 0.625i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 + (3.27 + 3.27i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.98 - 8.63i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.869 + 3.24i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-1.51 - 5.66i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-6.18 - 1.65i)T + (52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (1.60 + 5.97i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 5.14T + 71T^{2} \) |
| 73 | \( 1 + (-3.49 + 6.05i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-9.72 + 5.61i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (5.39 + 5.39i)T + 83iT^{2} \) |
| 89 | \( 1 + (-0.528 - 0.915i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 13.2iT - 97T^{2} \) |
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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
−10.71213330691894772909874988914, −9.478149836865871891324481753968, −8.870695223166534075872612023889, −7.53574448629786741884065269683, −6.64047712283970910422842797550, −5.93554378636820971898593427004, −5.29482871568398211813753677633, −4.46112004721130451745219106926, −3.53838625615133394451808785082, −0.826959857760939192956842353774,
0.951072544182422248981825877001, 2.08395881487329650818428639348, 3.89835334378665076952213129705, 4.67363765994731362850304055722, 5.71326224491725102129184275122, 6.45371571577851814958449643004, 6.97820545859048815552180016102, 8.653051817386292323910754625001, 9.653091265727487951914788145391, 10.60250938647444241852473620965