| L(s) = 1 | − 376·9-s + 1.95e3·11-s + 7.13e3·23-s − 4.86e3·25-s − 3.35e3·29-s − 9.20e3·37-s − 2.04e4·43-s − 1.02e5·53-s + 2.28e4·67-s + 1.53e5·71-s + 9.06e4·79-s + 5.30e4·81-s − 7.33e5·99-s + 2.01e5·107-s + 1.51e5·109-s − 7.26e4·113-s + 1.74e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2.24e5·169-s + ⋯ |
| L(s) = 1 | − 1.54·9-s + 4.86·11-s + 2.81·23-s − 1.55·25-s − 0.740·29-s − 1.10·37-s − 1.68·43-s − 5.03·53-s + 0.623·67-s + 3.62·71-s + 1.63·79-s + 0.897·81-s − 7.52·99-s + 1.70·107-s + 1.21·109-s − 0.535·113-s + 10.8·121-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 − 0.604·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(8.963729747\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.963729747\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 376 T^{2} + 88354 T^{4} + 376 p^{10} T^{6} + p^{20} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 4868 T^{2} + 22562806 T^{4} + 4868 p^{10} T^{6} + p^{20} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 976 T + 556178 T^{2} - 976 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 224500 T^{2} - 32848089674 T^{4} + 224500 p^{10} T^{6} + p^{20} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 3065760 T^{2} + 6374095942466 T^{4} + 3065760 p^{10} T^{6} + p^{20} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 8544504 T^{2} + 30052000046306 T^{4} + 8544504 p^{10} T^{6} + p^{20} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 3568 T + 12712350 T^{2} - 3568 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 1676 T + 24360510 T^{2} + 1676 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 29395756 T^{2} + 658228795954534 T^{4} + 29395756 p^{10} T^{6} + p^{20} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 4604 T + 143922030 T^{2} + 4604 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 212331360 T^{2} + 36724946155085474 T^{4} + 212331360 p^{10} T^{6} + p^{20} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 10224 T + 293018130 T^{2} + 10224 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 568199404 T^{2} + 172542701189040294 T^{4} + 568199404 p^{10} T^{6} + p^{20} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 51460 T + 1308627278 T^{2} + 51460 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 1609944152 T^{2} + 1391734325057519170 T^{4} + 1609944152 p^{10} T^{6} + p^{20} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 1094138468 T^{2} + 1609346124964468758 T^{4} + 1094138468 p^{10} T^{6} + p^{20} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 11448 T - 1089307338 T^{2} - 11448 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 76912 T + 4562060670 T^{2} - 76912 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 2170565248 T^{2} + 425171557315203874 T^{4} + 2170565248 p^{10} T^{6} + p^{20} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 45344 T + 6154499470 T^{2} - 45344 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 6721792472 T^{2} + 21939991980485194594 T^{4} + 6721792472 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 14718300768 T^{2} + \)\(11\!\cdots\!58\)\( T^{4} + 14718300768 p^{10} T^{6} + p^{20} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 22489794400 T^{2} + 25388976518139394 p^{2} T^{4} + 22489794400 p^{10} T^{6} + p^{20} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.71489036501561393027249325490, −6.20077268510865574526991915173, −6.12633707900141863610938475649, −6.10999532522848571207363246187, −6.06236846835353797851874477646, −5.14786779563350748363763830657, −5.10246190593881050153260233277, −5.08013532794258773703006378565, −4.95530097879747182709761881640, −4.34782564733914774789390977894, −4.05516040807204652221302834739, −3.90278441845350017878514977719, −3.74072775402439171627684324660, −3.37190374249083763231108961238, −3.30770949362189542206696523423, −3.00390087799909120649706369520, −2.81950507290415785181876466756, −2.07793518171309791100860104357, −1.91865051537295262688619234919, −1.67770159976970718883657849882, −1.48371166706497538348399988319, −1.21947282295432763668721103644, −0.75787750129704031880725959723, −0.56889770260486184472903104624, −0.30864521820180407176553164273,
0.30864521820180407176553164273, 0.56889770260486184472903104624, 0.75787750129704031880725959723, 1.21947282295432763668721103644, 1.48371166706497538348399988319, 1.67770159976970718883657849882, 1.91865051537295262688619234919, 2.07793518171309791100860104357, 2.81950507290415785181876466756, 3.00390087799909120649706369520, 3.30770949362189542206696523423, 3.37190374249083763231108961238, 3.74072775402439171627684324660, 3.90278441845350017878514977719, 4.05516040807204652221302834739, 4.34782564733914774789390977894, 4.95530097879747182709761881640, 5.08013532794258773703006378565, 5.10246190593881050153260233277, 5.14786779563350748363763830657, 6.06236846835353797851874477646, 6.10999532522848571207363246187, 6.12633707900141863610938475649, 6.20077268510865574526991915173, 6.71489036501561393027249325490
