| L(s) = 1 | + 64·4-s + 3.07e3·16-s − 2.29e4·19-s − 1.59e5·31-s + 2.08e3·49-s + 5.30e4·61-s + 1.31e5·64-s − 1.47e6·76-s + 1.40e5·79-s − 7.77e5·109-s + 3.49e6·121-s − 1.01e7·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 3.37e6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 4-s + 3/4·16-s − 3.34·19-s − 5.34·31-s + 0.0177·49-s + 0.233·61-s + 1/2·64-s − 3.34·76-s + 0.284·79-s − 0.600·109-s + 1.97·121-s − 5.34·124-s − 0.700·169-s + 0.0177·196-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.01492790582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01492790582\) |
| \(L(4)\) |
|
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 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 - 1042 T^{2} + p^{12} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 1745714 T^{2} + p^{12} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 1689806 T^{2} + p^{12} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 48274976 T^{2} + p^{12} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 5744 T + p^{6} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 284666690 T^{2} + p^{12} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 327940544 T^{2} + p^{12} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 39796 T + p^{6} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 2372472142 T^{2} + p^{12} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 8128061984 T^{2} + p^{12} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 12628286098 T^{2} + p^{12} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 15661450658 T^{2} + p^{12} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 12666935680 T^{2} + p^{12} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 21940729490 T^{2} + p^{12} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13250 T + p^{6} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 152366579314 T^{2} + p^{12} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 26256724990 T^{2} + p^{12} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 246904463842 T^{2} + p^{12} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 35116 T + p^{6} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 653760187346 T^{2} + p^{12} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 977236742720 T^{2} + p^{12} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 1562630622082 T^{2} + p^{12} T^{4} )^{2} \) |
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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
−7.02279147335497505629655492110, −6.82987913097687210742662742186, −6.64080699897648056910643126281, −6.16367077517693196373233537916, −5.88308183936313978088671266778, −5.81795119501360122838768672150, −5.81521161170613892319151878775, −5.28215013980460063455164564461, −4.84508188244835724964743012338, −4.78776845106633537829138911965, −4.56974238871984432466285183147, −3.91474859212138400031922609444, −3.78563951506468269261148262940, −3.70836332442545964810838426388, −3.53133474593172217759856947323, −3.01579393317100391610220840970, −2.49161484828858945990586874993, −2.38796834328966200326328669882, −2.15657632206220019972800821711, −1.86688445988452093051996070735, −1.62907784039014032415776711423, −1.33128022184041097542168907228, −0.913108965301475861497382379319, −0.28804060155400237694847609151, −0.02047993306568637606701044091,
0.02047993306568637606701044091, 0.28804060155400237694847609151, 0.913108965301475861497382379319, 1.33128022184041097542168907228, 1.62907784039014032415776711423, 1.86688445988452093051996070735, 2.15657632206220019972800821711, 2.38796834328966200326328669882, 2.49161484828858945990586874993, 3.01579393317100391610220840970, 3.53133474593172217759856947323, 3.70836332442545964810838426388, 3.78563951506468269261148262940, 3.91474859212138400031922609444, 4.56974238871984432466285183147, 4.78776845106633537829138911965, 4.84508188244835724964743012338, 5.28215013980460063455164564461, 5.81521161170613892319151878775, 5.81795119501360122838768672150, 5.88308183936313978088671266778, 6.16367077517693196373233537916, 6.64080699897648056910643126281, 6.82987913097687210742662742186, 7.02279147335497505629655492110
