| L(s) = 1 | + 107·4-s + 274·9-s − 752·11-s + 6.64e3·16-s + 624·19-s + 1.62e3·29-s + 1.54e4·31-s + 2.93e4·36-s + 1.56e4·41-s − 8.04e4·44-s + 2.49e4·49-s + 7.78e4·59-s + 3.96e3·61-s + 3.06e5·64-s + 1.34e5·71-s + 6.67e4·76-s + 1.10e5·79-s + 5.81e4·81-s − 2.33e5·89-s − 2.06e5·99-s + 2.72e5·101-s + 3.21e5·109-s + 1.73e5·116-s − 2.30e5·121-s + 1.65e6·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 3.34·4-s + 1.12·9-s − 1.87·11-s + 6.48·16-s + 0.396·19-s + 0.358·29-s + 2.88·31-s + 3.77·36-s + 1.45·41-s − 6.26·44-s + 1.48·49-s + 2.91·59-s + 0.136·61-s + 9.35·64-s + 3.17·71-s + 1.32·76-s + 1.99·79-s + 0.984·81-s − 3.11·89-s − 2.11·99-s + 2.65·101-s + 2.59·109-s + 1.19·116-s − 1.43·121-s + 9.64·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(29.37335863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(29.37335863\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 107 T^{2} + 1201 p^{2} T^{4} - 107 p^{10} T^{6} + p^{20} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 - 274 T^{2} + 16915 T^{4} - 274 p^{10} T^{6} + p^{20} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 24930 T^{2} + 693337523 T^{4} - 24930 p^{10} T^{6} + p^{20} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 376 T + 327458 T^{2} + 376 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2081714 T^{2} + 4633733514547 T^{4} - 2081714 p^{10} T^{6} + p^{20} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 312 T + 4973202 T^{2} - 312 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 5259708 T^{2} + 31095505529126 T^{4} - 5259708 p^{10} T^{6} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 28 p T + 41177342 T^{2} - 28 p^{6} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 7720 T + 71304910 T^{2} - 7720 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 106502146 T^{2} + 8363661311123827 T^{4} - 106502146 p^{10} T^{6} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 7840 T + 230885010 T^{2} - 7840 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 395899618 T^{2} + 81853058635202179 T^{4} - 395899618 p^{10} T^{6} + p^{20} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 72833986 T^{2} + 67005418961167347 T^{4} - 72833986 p^{10} T^{6} + p^{20} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 322170996 T^{2} - 1682173848365098 T^{4} - 322170996 p^{10} T^{6} + p^{20} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 38936 T + 1489531170 T^{2} - 38936 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 1984 T + 1322910298 T^{2} - 1984 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2955211300 T^{2} + 5828195697710504086 T^{4} - 2955211300 p^{10} T^{6} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 67396 T + 4686895793 T^{2} - 67396 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 2595687900 T^{2} + 2173198333277194598 T^{4} - 2595687900 p^{10} T^{6} + p^{20} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 55296 T + 6515668494 T^{2} - 55296 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 10160434700 T^{2} + 49017311554010412598 T^{4} - 10160434700 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 116508 T + 12213785846 T^{2} + 116508 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 4437044860 T^{2} + \)\(13\!\cdots\!98\)\( T^{4} - 4437044860 p^{10} T^{6} + p^{20} T^{8} \) |
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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
−7.78419810428153602987379726137, −6.93810032578067032237467931629, −6.93049140044499615373628670003, −6.90245888033848698057331926892, −6.88532191801024420497343492962, −6.29994817712723533104062909448, −6.11894128334142572857531253788, −5.75269199627176518762358878408, −5.53121559741602525414555352641, −5.45256091074438452250373882051, −4.94738980644687783394738579698, −4.55321703664035081166936128000, −4.46246025554232314099241058678, −3.87294194816134188849501904518, −3.56298270293050479251739301336, −3.21992134103559000022497634865, −2.99774593278565662128305017030, −2.51946607246138405239096836459, −2.25152139242695328861473464611, −2.22524847524303873398903522145, −2.17676151136482376055459836553, −1.28742711502708177271376046921, −1.06901923142887207122043397403, −0.862695194423678915949105045018, −0.48929009873980704891511492099,
0.48929009873980704891511492099, 0.862695194423678915949105045018, 1.06901923142887207122043397403, 1.28742711502708177271376046921, 2.17676151136482376055459836553, 2.22524847524303873398903522145, 2.25152139242695328861473464611, 2.51946607246138405239096836459, 2.99774593278565662128305017030, 3.21992134103559000022497634865, 3.56298270293050479251739301336, 3.87294194816134188849501904518, 4.46246025554232314099241058678, 4.55321703664035081166936128000, 4.94738980644687783394738579698, 5.45256091074438452250373882051, 5.53121559741602525414555352641, 5.75269199627176518762358878408, 6.11894128334142572857531253788, 6.29994817712723533104062909448, 6.88532191801024420497343492962, 6.90245888033848698057331926892, 6.93049140044499615373628670003, 6.93810032578067032237467931629, 7.78419810428153602987379726137
