| L(s) = 1 | + 4·3-s − 8·5-s + 4·7-s + 10·9-s + 4·11-s − 32·15-s + 12·19-s + 16·21-s + 12·23-s + 30·25-s + 20·27-s − 8·29-s + 16·31-s + 16·33-s − 32·35-s + 12·41-s + 12·43-s − 80·45-s + 8·49-s − 16·53-s − 32·55-s + 48·57-s − 4·59-s + 40·63-s + 12·67-s + 48·69-s − 12·71-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 3.57·5-s + 1.51·7-s + 10/3·9-s + 1.20·11-s − 8.26·15-s + 2.75·19-s + 3.49·21-s + 2.50·23-s + 6·25-s + 3.84·27-s − 1.48·29-s + 2.87·31-s + 2.78·33-s − 5.40·35-s + 1.87·41-s + 1.82·43-s − 11.9·45-s + 8/7·49-s − 2.19·53-s − 4.31·55-s + 6.35·57-s − 0.520·59-s + 5.03·63-s + 1.46·67-s + 5.77·69-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.677096146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.677096146\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 - 2 T^{2} + p^{2} T^{4} ) \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + 4 T + p T^{2} )^{2}( 1 + 8 T^{2} + p^{2} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 14 T^{2} + p^{2} T^{4} ) \) |
| 13 | $D_4\times C_2$ | \( 1 + 2 T^{2} - 48 T^{3} + 2 T^{4} - 48 p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 12 T + 86 T^{2} - 492 T^{3} + 2402 T^{4} - 492 p T^{5} + 86 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 300 T^{3} + 1246 T^{4} - 300 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 8 T + 34 T^{2} + 200 T^{3} + 1026 T^{4} + 200 p T^{5} + 34 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 + 2 T^{2} - 144 T^{3} + 2 T^{4} - 144 p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 516 T^{3} + 3694 T^{4} - 516 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 12 T + 38 T^{2} + 564 T^{3} - 6334 T^{4} + 564 p T^{5} + 38 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 486 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 16 T + 82 T^{2} - 128 T^{3} - 3294 T^{4} - 128 p T^{5} + 82 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 4 T + 22 T^{2} - 668 T^{3} - 2718 T^{4} - 668 p T^{5} + 22 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 2 T^{2} - 240 T^{3} + 2 T^{4} - 240 p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 12 T + 86 T^{2} - 876 T^{3} + 7010 T^{4} - 876 p T^{5} + 86 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 876 T^{3} + 10654 T^{4} + 876 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 4 T + 22 T^{2} + 1052 T^{3} - 4254 T^{4} + 1052 p T^{5} + 22 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 516 T^{3} + 1582 T^{4} + 516 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.897825510294918740728154580855, −7.61456102148798765254445019921, −7.51952781678018032241286921227, −7.35236703304040547598738008067, −7.28788736992689290156643864222, −6.93561384210194964130756690710, −6.76641026831060389585121358103, −6.20719131419109738494614294249, −5.99110483194666416004385926494, −5.53442029844512574071623305195, −5.10815283597588213893067251017, −4.88388952488504719873670699192, −4.71581868898017239878405779641, −4.25366720111004955444283823362, −4.09514314113447049274347440304, −4.05947621191648753863234279951, −3.92085760995649384641878433329, −3.10067551114759327246420915122, −3.09249657116822073639937590320, −3.02061048733000472834350914203, −2.86639213727961658223917803005, −2.03631312945521435935821488884, −1.48359960357213447278970742131, −1.03151084241967446087207868855, −0.921255956001779853244271660467,
0.921255956001779853244271660467, 1.03151084241967446087207868855, 1.48359960357213447278970742131, 2.03631312945521435935821488884, 2.86639213727961658223917803005, 3.02061048733000472834350914203, 3.09249657116822073639937590320, 3.10067551114759327246420915122, 3.92085760995649384641878433329, 4.05947621191648753863234279951, 4.09514314113447049274347440304, 4.25366720111004955444283823362, 4.71581868898017239878405779641, 4.88388952488504719873670699192, 5.10815283597588213893067251017, 5.53442029844512574071623305195, 5.99110483194666416004385926494, 6.20719131419109738494614294249, 6.76641026831060389585121358103, 6.93561384210194964130756690710, 7.28788736992689290156643864222, 7.35236703304040547598738008067, 7.51952781678018032241286921227, 7.61456102148798765254445019921, 7.897825510294918740728154580855
