L(s) = 1 | + (0.527 + 0.913i)2-s − 3-s + (0.443 − 0.768i)4-s + 0.832·5-s + (−0.527 − 0.913i)6-s + (1.13 − 1.95i)7-s + 3.04·8-s + 9-s + (0.439 + 0.760i)10-s + (−0.713 + 1.23i)11-s + (−0.443 + 0.768i)12-s + (0.308 + 0.535i)13-s + 2.38·14-s − 0.832·15-s + (0.718 + 1.24i)16-s + (2.57 + 4.46i)17-s + ⋯ |
L(s) = 1 | + (0.372 + 0.645i)2-s − 0.577·3-s + (0.221 − 0.384i)4-s + 0.372·5-s + (−0.215 − 0.372i)6-s + (0.427 − 0.739i)7-s + 1.07·8-s + 0.333·9-s + (0.138 + 0.240i)10-s + (−0.215 + 0.372i)11-s + (−0.128 + 0.221i)12-s + (0.0856 + 0.148i)13-s + 0.637·14-s − 0.215·15-s + (0.179 + 0.311i)16-s + (0.625 + 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.268i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 - 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.43778 + 0.196908i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.43778 + 0.196908i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 67 | \( 1 + (7.67 + 2.84i)T \) |
good | 2 | \( 1 + (-0.527 - 0.913i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 - 0.832T + 5T^{2} \) |
| 7 | \( 1 + (-1.13 + 1.95i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.713 - 1.23i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.308 - 0.535i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.57 - 4.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.939 + 1.62i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.55 + 2.69i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.962 - 1.66i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.286 - 0.495i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.67 + 4.63i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.25 - 7.36i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 7.16T + 43T^{2} \) |
| 47 | \( 1 + (-0.624 + 1.08i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 4.80T + 53T^{2} \) |
| 59 | \( 1 - 7.75T + 59T^{2} \) |
| 61 | \( 1 + (1.96 + 3.41i)T + (-30.5 + 52.8i)T^{2} \) |
| 71 | \( 1 + (4.64 - 8.05i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.88 - 13.6i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.745 - 1.29i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.355 + 0.616i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 0.514T + 89T^{2} \) |
| 97 | \( 1 + (-0.590 - 1.02i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.66135857806901598712729360475, −11.38890282678737049199447018499, −10.51957034365673642179401594434, −9.883816428013626607380392135079, −8.157574411713261248393233003599, −7.12586445297058036069078827403, −6.20988613533507674950079249491, −5.23575718416883969872403573802, −4.16055091844605923644520770839, −1.65598373386351520776530467591,
1.94963898123484189169981767258, 3.39885774991435842115899464408, 4.92926382415114839823674496285, 5.88950829992771066964229129334, 7.31794066406767509446388437883, 8.352450813406031258266079578036, 9.748729654866114880701003647776, 10.69819131390892445005506616784, 11.82360762970424889448237670542, 11.97378857427497894349028721175