| L(s) = 1 | − 540·5-s + 3.97e5·9-s − 8.77e6·13-s − 7.87e7·17-s − 4.88e8·25-s + 3.27e8·29-s + 7.20e9·37-s + 4.24e9·41-s − 2.14e8·45-s − 5.10e9·49-s − 2.71e10·53-s + 7.09e10·61-s + 4.73e9·65-s − 1.19e10·73-s − 1.24e11·81-s + 4.25e10·85-s − 1.50e12·89-s − 1.94e12·97-s + 2.01e12·101-s + 2.46e12·109-s + 6.04e12·113-s − 3.48e12·117-s + 3.54e12·121-s + 3.95e11·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.0345·5-s + 0.748·9-s − 1.81·13-s − 3.26·17-s − 1.99·25-s + 0.549·29-s + 2.80·37-s + 0.894·41-s − 0.0258·45-s − 0.368·49-s − 1.22·53-s + 1.37·61-s + 0.0628·65-s − 0.0790·73-s − 0.440·81-s + 0.112·85-s − 3.03·89-s − 2.33·97-s + 1.89·101-s + 1.47·109-s + 2.90·113-s − 1.35·117-s + 1.13·121-s + 0.103·125-s − 0.0190·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+6)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(1.040927870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.040927870\) |
| \(L(7)\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 14726 p^{3} T^{2} + p^{24} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 54 p T + p^{12} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 729297874 p T^{2} + p^{24} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 322617541222 p T^{2} + p^{24} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4386350 T + p^{12} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 39383550 T + p^{12} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 1482382886332322 T^{2} + p^{24} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 13940422220450878 T^{2} + p^{24} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 163560978 T + p^{12} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1521742293093899522 T^{2} + p^{24} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 3600024050 T + p^{12} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2124864738 T + p^{12} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 73135041050432481122 T^{2} + p^{24} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 76729470925866191998 T^{2} + p^{24} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 13585251150 T + p^{12} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - \)\(29\!\cdots\!62\)\( T^{2} + p^{24} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 35496554258 T + p^{12} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - \)\(14\!\cdots\!22\)\( T^{2} + p^{24} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - \)\(17\!\cdots\!82\)\( T^{2} + p^{24} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 5982269150 T + p^{12} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - \)\(10\!\cdots\!82\)\( T^{2} + p^{24} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - \)\(55\!\cdots\!02\)\( T^{2} + p^{24} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 753989286942 T + p^{12} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 970603845950 T + p^{12} T^{2} )^{2} \) |
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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
−16.45173641579911494963432989282, −15.54901918209113364478999916766, −15.42285661434096535589549387815, −14.53424765393267501546454662761, −13.71403536040581059576772715477, −13.00412984598497790545783145346, −12.57005163851365104825117544890, −11.42032499966870580239066793270, −11.14361209533333589836052312272, −9.771190881328347893332020980107, −9.655142482550109840001759991771, −8.506759102513945019756856288995, −7.55394315475801950121053792152, −6.87293260096239717638981542024, −5.98811764880138173666071788211, −4.46653623063737179605744498951, −4.39878940698416530283590930981, −2.56132212862914461330099792769, −1.98842836974668574194191951257, −0.39631622499006013507911395808,
0.39631622499006013507911395808, 1.98842836974668574194191951257, 2.56132212862914461330099792769, 4.39878940698416530283590930981, 4.46653623063737179605744498951, 5.98811764880138173666071788211, 6.87293260096239717638981542024, 7.55394315475801950121053792152, 8.506759102513945019756856288995, 9.655142482550109840001759991771, 9.771190881328347893332020980107, 11.14361209533333589836052312272, 11.42032499966870580239066793270, 12.57005163851365104825117544890, 13.00412984598497790545783145346, 13.71403536040581059576772715477, 14.53424765393267501546454662761, 15.42285661434096535589549387815, 15.54901918209113364478999916766, 16.45173641579911494963432989282
