| L(s) = 1 | − 23·4-s + 38·7-s − 604·13-s + 256·16-s − 1.21e3·19-s − 161·25-s − 874·28-s − 478·31-s + 2.96e3·37-s + 1.96e3·43-s + 5.16e3·49-s + 1.38e4·52-s + 632·61-s − 5.49e3·64-s − 9.24e3·67-s − 1.21e4·73-s + 2.79e4·76-s + 2.09e4·79-s − 2.29e4·91-s + 1.30e4·97-s + 3.70e3·100-s + 1.53e4·103-s + 9.29e3·109-s + 9.72e3·112-s − 1.41e4·121-s + 1.09e4·124-s + 127-s + ⋯ |
| L(s) = 1 | − 1.43·4-s + 0.775·7-s − 3.57·13-s + 16-s − 3.36·19-s − 0.257·25-s − 1.11·28-s − 0.497·31-s + 2.16·37-s + 1.06·43-s + 2.15·49-s + 5.13·52-s + 0.169·61-s − 1.34·64-s − 2.05·67-s − 2.27·73-s + 4.84·76-s + 3.34·79-s − 2.77·91-s + 1.38·97-s + 0.370·100-s + 1.44·103-s + 0.782·109-s + 0.775·112-s − 0.966·121-s + 0.715·124-s + 6.20e−5·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43046721 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43046721 ^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.5345569637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5345569637\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 + 23 T^{2} + 273 T^{4} + 23 p^{8} T^{6} + p^{16} T^{8} \) |
| 5 | $C_2^3$ | \( 1 + 161 T^{2} - 364704 T^{4} + 161 p^{8} T^{6} + p^{16} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 19 T - 2040 T^{2} - 19 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 14153 T^{2} - 14051472 T^{4} + 14153 p^{8} T^{6} + p^{16} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 302 T + 62643 T^{2} + 302 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 4354 T^{2} + p^{8} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 16 p T + p^{4} T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 469682 T^{2} + 142290195843 T^{4} + 469682 p^{8} T^{6} + p^{16} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 954878 T^{2} + 411545581923 T^{4} + 954878 p^{8} T^{6} + p^{16} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 239 T - 866400 T^{2} + 239 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 20 p T + p^{4} T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 + 5599538 T^{2} + 23369900584323 T^{4} + 5599538 p^{8} T^{6} + p^{16} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 982 T - 2454477 T^{2} - 982 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 5067806 T^{2} + 1871370991875 T^{4} + 5067806 p^{8} T^{6} + p^{16} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 13243313 T^{2} + p^{8} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 15696638 T^{2} + 99554006898723 T^{4} + 15696638 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 316 T - 13745985 T^{2} - 316 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 4622 T + 1211763 T^{2} + 4622 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 47518238 T^{2} + p^{8} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 3031 T + p^{4} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 10450 T + 70252419 T^{2} - 10450 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 64676047 T^{2} + 1930698823407168 T^{4} - 64676047 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 76456478 T^{2} + p^{8} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 6517 T - 46057992 T^{2} - 6517 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
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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
−9.719078248340046644207120231587, −9.403675541903555486963959011721, −9.369547311266359727078860977385, −9.158161750367782391568243143488, −8.556770352453322355553723334713, −8.309842030103785920579262106975, −8.287083217550770834161356139035, −7.66985850268917462170401229173, −7.34322715366949988407670569199, −7.25073189644504571120553456566, −6.96455927262698838135621583620, −6.11084098408311875549062108965, −5.93090228531331700338039955874, −5.84052999462538836234084606859, −4.94200794977494878192124117472, −4.79466649177512772234651415063, −4.58442461125484109628156794585, −4.33624709937230634221049811947, −4.04100070177196943682835271990, −3.24706856739956631434799988273, −2.55407982440465036166271079026, −2.18592509392802626741326275278, −2.00462087242337547024961091671, −0.71674293012384824469758277552, −0.25279443193821078283614096466,
0.25279443193821078283614096466, 0.71674293012384824469758277552, 2.00462087242337547024961091671, 2.18592509392802626741326275278, 2.55407982440465036166271079026, 3.24706856739956631434799988273, 4.04100070177196943682835271990, 4.33624709937230634221049811947, 4.58442461125484109628156794585, 4.79466649177512772234651415063, 4.94200794977494878192124117472, 5.84052999462538836234084606859, 5.93090228531331700338039955874, 6.11084098408311875549062108965, 6.96455927262698838135621583620, 7.25073189644504571120553456566, 7.34322715366949988407670569199, 7.66985850268917462170401229173, 8.287083217550770834161356139035, 8.309842030103785920579262106975, 8.556770352453322355553723334713, 9.158161750367782391568243143488, 9.369547311266359727078860977385, 9.403675541903555486963959011721, 9.719078248340046644207120231587
