| L(s) = 1 | + (−1.57 + 0.724i)3-s + (1.94 − 2.28i)9-s − 6.61i·11-s + 2.36·17-s − 8.34·19-s + (−1.41 + 5.00i)27-s + (4.79 + 10.3i)33-s + 0.460i·41-s + 10i·43-s − 7·49-s + (−3.72 + 1.71i)51-s + (13.1 − 6.05i)57-s + 14.1i·59-s − 14.3i·67-s + 13.6i·73-s + ⋯ |
| L(s) = 1 | + (−0.908 + 0.418i)3-s + (0.649 − 0.760i)9-s − 1.99i·11-s + 0.574·17-s − 1.91·19-s + (−0.272 + 0.962i)27-s + (0.833 + 1.81i)33-s + 0.0719i·41-s + 1.52i·43-s − 49-s + (−0.521 + 0.240i)51-s + (1.73 − 0.801i)57-s + 1.84i·59-s − 1.75i·67-s + 1.60i·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0319i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08304939242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08304939242\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.57 - 0.724i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 6.61iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 2.36T + 17T^{2} \) |
| 19 | \( 1 + 8.34T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 0.460iT - 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 14.1iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 14.3iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 13.6iT - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + 12.7iT - 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−8.583463561587991176512171200813, −7.951081418275001591187750434987, −6.73456779416261214030677473261, −6.11921465060296673122566232811, −5.60083732265888438148861432556, −4.58676897952497331191535240185, −3.78556967230874404445615896251, −2.86553092991111892812614172782, −1.24717418008420296692210151922, −0.03341534566370987926652358876,
1.62401183801837787451852164373, 2.33116980264953798994147476748, 3.97640931768406617251789032326, 4.66797087074622517353026846596, 5.39599357811044935962037602321, 6.41340660660268195265553598492, 6.93099237109056550554774744767, 7.64421385808327175117506039067, 8.453376923941937034871858496718, 9.532123035298096563669769895684
