L(s) = 1 | + 90·5-s + 504·11-s + 440·19-s + 4.97e3·25-s + 1.38e4·29-s − 1.35e4·31-s + 396·41-s + 3.00e4·49-s + 4.53e4·55-s − 4.93e4·59-s − 1.13e4·61-s + 1.06e5·71-s − 1.03e5·79-s + 1.99e4·89-s + 3.96e4·95-s + 2.18e5·101-s − 4.20e4·109-s − 1.31e5·121-s + 1.66e5·125-s + 127-s + 131-s + 137-s + 139-s + 1.24e6·145-s + 149-s + 151-s − 1.21e6·155-s + ⋯ |
L(s) = 1 | + 1.60·5-s + 1.25·11-s + 0.279·19-s + 1.59·25-s + 3.06·29-s − 2.52·31-s + 0.0367·41-s + 1.78·49-s + 2.02·55-s − 1.84·59-s − 0.392·61-s + 2.51·71-s − 1.87·79-s + 0.267·89-s + 0.450·95-s + 2.12·101-s − 0.338·109-s − 0.817·121-s + 0.953·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 4.92·145-s + 3.69e−6·149-s + 3.56e−6·151-s − 4.06·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(7.124533640\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.124533640\) |
\(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 - 18 p T + p^{5} T^{2} \) |
good | 7 | $C_2^2$ | \( 1 - 30050 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 252 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 728330 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2363810 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 220 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6946370 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6930 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6752 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 56462470 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 198 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 293842250 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 347593490 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 802472090 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 24660 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 5698 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 795787610 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 53352 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 883886830 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 51920 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 4053674810 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9990 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 6923133890 T^{2} + p^{10} T^{4} \) |
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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
−9.700666523348687745043247813654, −9.509824879243594892779965412343, −8.946115008338592699635025416803, −8.879766078992221920969633669209, −8.256038195068743085480254198922, −7.67347178334648190997557781558, −7.00659471411899864195927974180, −6.84166133765982907924657219422, −6.17321540506611517345470524712, −6.06745389451677500228793340953, −5.36520414027103962772436912856, −5.09067116896755134092841099086, −4.35770946052570680965135261146, −4.00282453510819375127940587782, −3.11554473428117728236681126926, −2.88818867417723993043865758414, −1.89354746575714583441968555692, −1.84908026162384818924121049253, −1.00945538302512885674216601645, −0.62026114143845586746120146031,
0.62026114143845586746120146031, 1.00945538302512885674216601645, 1.84908026162384818924121049253, 1.89354746575714583441968555692, 2.88818867417723993043865758414, 3.11554473428117728236681126926, 4.00282453510819375127940587782, 4.35770946052570680965135261146, 5.09067116896755134092841099086, 5.36520414027103962772436912856, 6.06745389451677500228793340953, 6.17321540506611517345470524712, 6.84166133765982907924657219422, 7.00659471411899864195927974180, 7.67347178334648190997557781558, 8.256038195068743085480254198922, 8.879766078992221920969633669209, 8.946115008338592699635025416803, 9.509824879243594892779965412343, 9.700666523348687745043247813654