| L(s) = 1 | + 24·17-s − 16·49-s − 24·53-s + 32·61-s − 9·81-s − 16·109-s + 24·113-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 5.82·17-s − 2.28·49-s − 3.29·53-s + 4.09·61-s − 81-s − 1.53·109-s + 2.25·113-s + 4/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.38·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.856372879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.856372879\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 112 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.96286599494741860273561082632, −6.56373813522172671280857867397, −6.54238952540630686659318318380, −6.53826806688443484105150056407, −5.94465182253247385241349778986, −5.72130294843633234431926905971, −5.55090204027147250939065129680, −5.46293762174267451844887393495, −5.34134021971849280562829356947, −5.08141218847295693075234329865, −4.65964101672166786085914853699, −4.59038584245134141689728967717, −4.05820026651018439650738125514, −3.99150396075454606348095535751, −3.52101798879906307408989502852, −3.39812385153242594565528537706, −3.12310875135830545435174305429, −3.02754418150256193495200735190, −2.93856843572055286182055090923, −2.20013094030594625422467046276, −1.93652270288700147074454678191, −1.45371600830483743795150592584, −1.35688950941579948201508818312, −0.902389848616965654331420979902, −0.54377063936868461743456342907,
0.54377063936868461743456342907, 0.902389848616965654331420979902, 1.35688950941579948201508818312, 1.45371600830483743795150592584, 1.93652270288700147074454678191, 2.20013094030594625422467046276, 2.93856843572055286182055090923, 3.02754418150256193495200735190, 3.12310875135830545435174305429, 3.39812385153242594565528537706, 3.52101798879906307408989502852, 3.99150396075454606348095535751, 4.05820026651018439650738125514, 4.59038584245134141689728967717, 4.65964101672166786085914853699, 5.08141218847295693075234329865, 5.34134021971849280562829356947, 5.46293762174267451844887393495, 5.55090204027147250939065129680, 5.72130294843633234431926905971, 5.94465182253247385241349778986, 6.53826806688443484105150056407, 6.54238952540630686659318318380, 6.56373813522172671280857867397, 6.96286599494741860273561082632
