| L(s) = 1 | + 60·4-s + 296·11-s + 2.57e3·16-s − 2.12e3·19-s − 6.82e3·29-s − 4.89e3·31-s + 1.87e4·41-s + 1.77e4·44-s − 3.25e3·49-s − 4.00e4·59-s + 6.46e4·61-s + 9.31e4·64-s + 6.52e4·71-s − 1.27e5·76-s + 6.67e4·79-s + 2.02e5·89-s + 1.79e5·101-s − 7.36e4·109-s − 4.09e5·116-s − 2.56e5·121-s − 2.93e5·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 15/8·4-s + 0.737·11-s + 2.51·16-s − 1.34·19-s − 1.50·29-s − 0.915·31-s + 1.74·41-s + 1.38·44-s − 0.193·49-s − 1.49·59-s + 2.22·61-s + 2.84·64-s + 1.53·71-s − 2.52·76-s + 1.20·79-s + 2.71·89-s + 1.75·101-s − 0.593·109-s − 2.82·116-s − 1.59·121-s − 1.71·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.967075778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.967075778\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 15 p^{2} T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 3250 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 148 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3910 p^{2} T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 24030 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 1060 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 4016110 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3410 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2448 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 138654790 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 9398 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 292469350 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 312570270 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 267759270 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 20020 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 32302 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 1017334570 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 32648 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2642720110 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 33360 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 7598656630 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 101370 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 3004635070 T^{2} + p^{10} T^{4} \) |
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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
−11.52754875252598540989743484064, −11.10405029313746446887448562810, −10.78708055024353268010157070845, −10.44596667286359341322326001092, −9.654759759568846321968885184156, −9.261877394176351524991132880577, −8.658584359998800304854877577487, −7.905920971567429481788373155079, −7.58428022467062287266018046946, −7.06749468039505220342846903490, −6.42498860151287064232203569582, −6.23799340076894017166890343514, −5.60869914040530507862078919076, −4.89077936787222148762140689743, −3.78157151679034175273047986240, −3.65966655599419577346465504890, −2.54542137291725847544868945281, −2.13731800534778857489148889745, −1.53671309530455615790778619244, −0.62298110197074692411003408939,
0.62298110197074692411003408939, 1.53671309530455615790778619244, 2.13731800534778857489148889745, 2.54542137291725847544868945281, 3.65966655599419577346465504890, 3.78157151679034175273047986240, 4.89077936787222148762140689743, 5.60869914040530507862078919076, 6.23799340076894017166890343514, 6.42498860151287064232203569582, 7.06749468039505220342846903490, 7.58428022467062287266018046946, 7.905920971567429481788373155079, 8.658584359998800304854877577487, 9.261877394176351524991132880577, 9.654759759568846321968885184156, 10.44596667286359341322326001092, 10.78708055024353268010157070845, 11.10405029313746446887448562810, 11.52754875252598540989743484064
