| L(s) = 1 | + 12·7-s − 4·13-s − 4·25-s + 24·31-s − 24·37-s + 24·43-s + 58·49-s − 4·61-s + 12·67-s + 24·73-s − 36·79-s − 48·91-s − 28·97-s − 108·103-s + 48·109-s + 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 34·169-s + 173-s + ⋯ |
| L(s) = 1 | + 4.53·7-s − 1.10·13-s − 4/5·25-s + 4.31·31-s − 3.94·37-s + 3.65·43-s + 58/7·49-s − 0.512·61-s + 1.46·67-s + 2.80·73-s − 4.05·79-s − 5.03·91-s − 2.84·97-s − 10.6·103-s + 4.59·109-s + 2.54·121-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 + 2.61·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.742455371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.742455371\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 4 T^{2} + 2 p T^{4} - 176 T^{6} - 989 T^{8} - 176 p^{2} T^{10} + 2 p^{5} T^{12} + 4 p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( ( 1 - 6 T + 25 T^{2} - 78 T^{3} + 204 T^{4} - 78 p T^{5} + 25 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 - 28 T^{2} + 394 T^{4} - 4144 T^{6} + 42595 T^{8} - 4144 p^{2} T^{10} + 394 p^{4} T^{12} - 28 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( ( 1 + 2 T - 11 T^{2} - 22 T^{3} + 4 T^{4} - 22 p T^{5} - 11 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 20 T^{2} + 246 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( 1 + 20 T^{2} - 326 T^{4} - 6640 T^{6} + 35635 T^{8} - 6640 p^{2} T^{10} - 326 p^{4} T^{12} + 20 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( 1 + 52 T^{2} + 1114 T^{4} - 4784 T^{6} - 453245 T^{8} - 4784 p^{2} T^{10} + 1114 p^{4} T^{12} + 52 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 - 12 T + 106 T^{2} - 696 T^{3} + 3891 T^{4} - 696 p T^{5} + 106 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 6 T + 71 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( 1 + 100 T^{2} + 4906 T^{4} + 173200 T^{6} + 6313075 T^{8} + 173200 p^{2} T^{10} + 4906 p^{4} T^{12} + 100 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 - 12 T + 130 T^{2} - 984 T^{3} + 6939 T^{4} - 984 p T^{5} + 130 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 76 T^{2} + 1114 T^{4} - 18544 T^{6} + 4095379 T^{8} - 18544 p^{2} T^{10} + 1114 p^{4} T^{12} - 76 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 20 T^{2} - 1194 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 28 T^{2} - 4022 T^{4} + 60368 T^{6} + 8251171 T^{8} + 60368 p^{2} T^{10} - 4022 p^{4} T^{12} - 28 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 + 2 T - 11 T^{2} - 214 T^{3} - 3740 T^{4} - 214 p T^{5} - 11 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 6 T + 85 T^{2} - 438 T^{3} + 1644 T^{4} - 438 p T^{5} + 85 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 6 T + 107 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 18 T + 193 T^{2} + 1530 T^{3} + 9516 T^{4} + 1530 p T^{5} + 193 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 268 T^{2} + 40858 T^{4} - 4606384 T^{6} + 416639299 T^{8} - 4606384 p^{2} T^{10} + 40858 p^{4} T^{12} - 268 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 308 T^{2} + 39126 T^{4} - 308 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 7 T - 48 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.72848586742577360610705094838, −3.62290105400001715203065017483, −3.55962063710939476182441538773, −3.32635980458357065461916178577, −3.19221474523152185786104046603, −2.90798847464328010603789018021, −2.87619829618007826494505335609, −2.78521930363225218992522311452, −2.75653642724655033513766547407, −2.65064015558349915867543507966, −2.42706204143799214884577337473, −2.18881452094541653214582722253, −2.15171726885582872712061999835, −2.03984795004479101269251531515, −2.03438161207632967373953519276, −1.75117236864064012898646285827, −1.51874831619176091060084567020, −1.47787668837549908216221731891, −1.32282276629635955326499121004, −1.28626626205611577494245615879, −1.16496040681687428794552693629, −0.869576036053740533631044052665, −0.72919316802352239534181730544, −0.42723872163521850457252646256, −0.07439196177467973919082045760,
0.07439196177467973919082045760, 0.42723872163521850457252646256, 0.72919316802352239534181730544, 0.869576036053740533631044052665, 1.16496040681687428794552693629, 1.28626626205611577494245615879, 1.32282276629635955326499121004, 1.47787668837549908216221731891, 1.51874831619176091060084567020, 1.75117236864064012898646285827, 2.03438161207632967373953519276, 2.03984795004479101269251531515, 2.15171726885582872712061999835, 2.18881452094541653214582722253, 2.42706204143799214884577337473, 2.65064015558349915867543507966, 2.75653642724655033513766547407, 2.78521930363225218992522311452, 2.87619829618007826494505335609, 2.90798847464328010603789018021, 3.19221474523152185786104046603, 3.32635980458357065461916178577, 3.55962063710939476182441538773, 3.62290105400001715203065017483, 3.72848586742577360610705094838
Plot not available for L-functions of degree greater than 10.
