| L(s) = 1 | + 26·9-s − 144·11-s − 250·25-s + 108·29-s + 3.31e3·71-s + 472·79-s + 729·81-s − 3.74e3·99-s + 4.53e3·109-s + 7.84e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2.77e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 0.962·9-s − 3.94·11-s − 2·25-s + 0.691·29-s + 5.53·71-s + 0.672·79-s + 81-s − 3.80·99-s + 3.98·109-s + 5.89·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 1.26·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.586446819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.586446819\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 - 26 T^{2} - 53 T^{4} - 26 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 72 T + 3853 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 + 2774 T^{2} + 2868267 T^{4} + 2774 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $C_2^3$ | \( 1 - 754 T^{2} - 23569053 T^{4} - 754 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 54 T - 21473 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 + 175646 T^{2} + 20072301987 T^{4} + 175646 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 - 828 T + p^{3} T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 504254 T^{2} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 236 T - 437343 T^{2} - 236 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 1141306 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 97 | $C_2^3$ | \( 1 - 897874 T^{2} - 26794285053 T^{4} - 897874 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−6.64192565423828222747694804729, −6.53636570672767498737907570578, −6.41404324270514775804568726030, −5.96160340350052057779627547131, −5.93044083086341170623503290120, −5.38743450941739229935309009624, −5.37964589920140539419063407625, −5.13383809617027936113538067054, −5.08113296879438934832771329549, −4.75476322047255846294178994227, −4.51987206003904552208199570212, −4.10563313234673694765069243828, −3.88934117605066799976935588722, −3.79638777177617859038963716748, −3.20282016697113893126024953806, −3.04900953813856112881261170646, −2.99320197640946981801198253894, −2.33498930810521376472879400489, −2.15266257992037944507869857084, −2.14562328032123788977757055094, −1.91538546984712603906558257126, −1.21226719389357184111707735085, −0.811728253828860352771716918708, −0.45767307402593772994276923120, −0.30019983986362418274967910728,
0.30019983986362418274967910728, 0.45767307402593772994276923120, 0.811728253828860352771716918708, 1.21226719389357184111707735085, 1.91538546984712603906558257126, 2.14562328032123788977757055094, 2.15266257992037944507869857084, 2.33498930810521376472879400489, 2.99320197640946981801198253894, 3.04900953813856112881261170646, 3.20282016697113893126024953806, 3.79638777177617859038963716748, 3.88934117605066799976935588722, 4.10563313234673694765069243828, 4.51987206003904552208199570212, 4.75476322047255846294178994227, 5.08113296879438934832771329549, 5.13383809617027936113538067054, 5.37964589920140539419063407625, 5.38743450941739229935309009624, 5.93044083086341170623503290120, 5.96160340350052057779627547131, 6.41404324270514775804568726030, 6.53636570672767498737907570578, 6.64192565423828222747694804729
