| L(s) = 1 | + 110·5-s − 1.00e3·11-s − 2.68e3·19-s + 8.97e3·25-s − 5.29e3·29-s + 1.12e4·31-s + 3.79e4·41-s + 3.35e4·49-s − 1.10e5·55-s + 5.66e4·59-s + 3.65e4·61-s − 5.76e4·71-s + 1.20e5·79-s + 4.53e4·89-s − 2.95e5·95-s − 3.35e5·101-s + 1.07e5·109-s + 4.27e5·121-s + 6.43e5·125-s + 127-s + 131-s + 137-s + 139-s − 5.82e5·145-s + 149-s + 151-s + 1.23e6·155-s + ⋯ |
| L(s) = 1 | + 1.96·5-s − 2.49·11-s − 1.70·19-s + 2.87·25-s − 1.16·29-s + 2.09·31-s + 3.52·41-s + 1.99·49-s − 4.90·55-s + 2.11·59-s + 1.25·61-s − 1.35·71-s + 2.17·79-s + 0.606·89-s − 3.36·95-s − 3.27·101-s + 0.867·109-s + 2.65·121-s + 3.68·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s − 2.29·145-s + 3.69e−6·149-s + 3.56e−6·151-s + 4.12·155-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
| \(L(3)\) |
\(\approx\) |
\(5.390668030\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.390668030\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 - 22 p T + p^{5} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 33598 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 500 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 659642 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 541458 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 1344 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3937314 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2646 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5612 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 85572970 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 18986 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 288237670 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 379480014 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 747967430 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 28300 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 18290 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 1649943722 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 28800 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3197010322 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 60228 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 7871990262 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 22678 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 15808047490 T^{2} + p^{10} T^{4} \) |
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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
−9.870336688405155462033196033545, −9.540458870487694395300064259884, −9.078077093933950672159320907466, −8.609137764800264835456201115155, −8.107569897613492255549136308167, −7.87464266792861300427478796728, −7.10909930857722787393161259337, −6.83889315816418784470008467093, −6.05489603642641213182137360567, −5.94637505827390992288057143877, −5.31889058284927699686812506812, −5.26564638364668762197672391066, −4.32608676442173408033883569061, −4.18383958594029829078765163134, −2.83647755418166033097872403944, −2.77026127161829561322939346156, −2.07900925256954359702997241864, −2.07654885080261927482334681499, −0.75398859597034106508219486324, −0.64532435682918592109578917605,
0.64532435682918592109578917605, 0.75398859597034106508219486324, 2.07654885080261927482334681499, 2.07900925256954359702997241864, 2.77026127161829561322939346156, 2.83647755418166033097872403944, 4.18383958594029829078765163134, 4.32608676442173408033883569061, 5.26564638364668762197672391066, 5.31889058284927699686812506812, 5.94637505827390992288057143877, 6.05489603642641213182137360567, 6.83889315816418784470008467093, 7.10909930857722787393161259337, 7.87464266792861300427478796728, 8.107569897613492255549136308167, 8.609137764800264835456201115155, 9.078077093933950672159320907466, 9.540458870487694395300064259884, 9.870336688405155462033196033545
