| L(s) = 1 | + 288·11-s − 280·19-s − 6.25e3·29-s − 1.03e4·31-s − 2.51e4·41-s − 2.59e4·49-s + 5.59e4·59-s + 4.40e4·61-s + 1.19e5·71-s − 2.20e4·79-s + 1.47e5·89-s + 2.24e5·101-s − 2.50e5·109-s − 2.59e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 7.40e5·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 0.717·11-s − 0.177·19-s − 1.38·29-s − 1.93·31-s − 2.33·41-s − 1.54·49-s + 2.09·59-s + 1.51·61-s + 2.80·71-s − 0.398·79-s + 1.97·89-s + 2.18·101-s − 2.02·109-s − 1.61·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 1.99·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.618474583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.618474583\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 25922 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 144 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 740086 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 823682 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 140 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 12483310 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3126 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 5176 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 107739290 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 12570 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 161398630 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 256123682 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 469362022 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 27984 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 22022 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2539569238 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 59520 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 356535218 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 11048 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 5412319370 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 73650 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 15912868030 T^{2} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.723947535120714606749606429788, −9.179167443417862889271686778887, −8.617399021823867877968507910178, −8.535599067821567635661911461863, −7.81642278033110949535767768222, −7.51921661751799046882452617923, −6.93934317517852293358999716506, −6.61535396397257729842599296933, −6.29043025082404779843484496274, −5.54987541178757266817364989985, −5.13442730902523155012294521472, −5.00696232642344119300475794303, −3.98819102672020661143484894118, −3.70880488728639216969221820674, −3.49881141042459606377398944846, −2.62529769469732662419386447848, −1.86632405063102128257544510814, −1.80212664117290811002340796346, −0.914328091855572895987537552650, −0.28145063330173351877594530065,
0.28145063330173351877594530065, 0.914328091855572895987537552650, 1.80212664117290811002340796346, 1.86632405063102128257544510814, 2.62529769469732662419386447848, 3.49881141042459606377398944846, 3.70880488728639216969221820674, 3.98819102672020661143484894118, 5.00696232642344119300475794303, 5.13442730902523155012294521472, 5.54987541178757266817364989985, 6.29043025082404779843484496274, 6.61535396397257729842599296933, 6.93934317517852293358999716506, 7.51921661751799046882452617923, 7.81642278033110949535767768222, 8.535599067821567635661911461863, 8.617399021823867877968507910178, 9.179167443417862889271686778887, 9.723947535120714606749606429788
