| L(s) = 1 | + 2.78i·2-s + (0.383 + 1.68i)3-s − 5.73·4-s + (−4.69 + 1.06i)6-s + 4.37·7-s − 10.3i·8-s + (−2.70 + 1.29i)9-s + (−2.19 − 9.68i)12-s + 12.1i·14-s + 17.3·16-s − 0.500i·17-s + (−3.59 − 7.52i)18-s + (1.67 + 7.39i)21-s + (17.5 − 3.97i)24-s − 5·25-s + ⋯ |
| L(s) = 1 | + 1.96i·2-s + (0.221 + 0.975i)3-s − 2.86·4-s + (−1.91 + 0.435i)6-s + 1.65·7-s − 3.66i·8-s + (−0.902 + 0.431i)9-s + (−0.634 − 2.79i)12-s + 3.25i·14-s + 4.34·16-s − 0.121i·17-s + (−0.848 − 1.77i)18-s + (0.366 + 1.61i)21-s + (3.57 − 0.811i)24-s − 25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.119298 - 1.06508i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.119298 - 1.06508i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.383 - 1.68i)T \) |
| 47 | \( 1 + 6.85iT \) |
| good | 2 | \( 1 - 2.78iT - 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 4.37T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 0.500iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 9.76T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 53 | \( 1 - 3.35iT - 53T^{2} \) |
| 59 | \( 1 - 2.20iT - 59T^{2} \) |
| 61 | \( 1 - 7.48T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 9.13iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 17.2T + 79T^{2} \) |
| 83 | \( 1 + 13.7iT - 83T^{2} \) |
| 89 | \( 1 + 14.4iT - 89T^{2} \) |
| 97 | \( 1 - 1.27T + 97T^{2} \) |
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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
−14.24413491068776774766030697665, −13.34986752094158615515769826928, −11.59719913749344125575497730975, −10.21279317516763703873443552792, −9.110863244603247173956826317827, −8.249721642516496517550066062627, −7.53980656729489232814184421143, −5.89212816809948154763421957570, −4.95624687896884213190649527767, −4.11466336690578390443742235904,
1.34984171508662701955400187784, 2.48695076226044251315487212157, 4.17089363349496733271319978210, 5.48591559019998655760935235055, 7.84337368244147036313551383651, 8.503137985724122809082873555074, 9.689260725266690168678549552692, 11.08596405371180034630164154285, 11.49667442724012270391892490428, 12.40496565889014563887670144656
