| L(s) = 1 | + 10·5-s − 124·13-s + 92·17-s + 75·25-s + 180·29-s − 428·37-s + 20·41-s + 294·49-s + 1.35e3·53-s + 500·61-s − 1.24e3·65-s + 1.04e3·73-s + 920·85-s − 1.94e3·89-s − 1.86e3·97-s + 1.20e3·101-s + 4.30e3·109-s + 4.36e3·113-s − 2.58e3·121-s + 500·125-s + 127-s + 131-s + 137-s + 139-s + 1.80e3·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 2.64·13-s + 1.31·17-s + 3/5·25-s + 1.15·29-s − 1.90·37-s + 0.0761·41-s + 6/7·49-s + 3.51·53-s + 1.04·61-s − 2.36·65-s + 1.67·73-s + 1.17·85-s − 2.31·89-s − 1.95·97-s + 1.18·101-s + 3.78·109-s + 3.63·113-s − 1.93·121-s + 0.357·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 1.03·145-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.844393682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.844393682\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 6 p^{2} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2582 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 62 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 46 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 2198 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 12646 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 90 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 36462 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 214 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 154514 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 49226 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 678 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 241478 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 250 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 599106 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 581342 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 522 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 217758 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 999074 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 970 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 934 T + p^{3} T^{2} )^{2} \) |
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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
−9.558022948169763498860957028814, −8.983627928779647567816298626939, −8.427838331288779887210326280015, −8.417994985471116099384001005169, −7.54135544425348171218500778432, −7.34620666484450842011991198076, −6.89884173275173622235119617998, −6.77719653018365708186155486420, −5.85726824639262399200624828530, −5.64104990070064938606814840571, −5.18700649392820428843871395614, −4.96114164849100291571240614971, −4.35559868509634142516369702348, −3.83630600896124833998605354063, −3.08895946567299908889273426666, −2.79402996792555998509872273302, −2.07325714783754065014939677866, −1.97741141288338885018042808299, −0.867507139521931367981835595353, −0.54307365479155893514727022506,
0.54307365479155893514727022506, 0.867507139521931367981835595353, 1.97741141288338885018042808299, 2.07325714783754065014939677866, 2.79402996792555998509872273302, 3.08895946567299908889273426666, 3.83630600896124833998605354063, 4.35559868509634142516369702348, 4.96114164849100291571240614971, 5.18700649392820428843871395614, 5.64104990070064938606814840571, 5.85726824639262399200624828530, 6.77719653018365708186155486420, 6.89884173275173622235119617998, 7.34620666484450842011991198076, 7.54135544425348171218500778432, 8.417994985471116099384001005169, 8.427838331288779887210326280015, 8.983627928779647567816298626939, 9.558022948169763498860957028814
