| L(s) = 1 | − 4·25-s + 32·37-s − 32·43-s + 80·67-s + 56·79-s + 80·109-s + 52·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 80·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 4/5·25-s + 5.26·37-s − 4.87·43-s + 9.77·67-s + 6.30·79-s + 7.66·109-s + 4.72·121-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 + 6.15·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.09488978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.09488978\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 + 2 T^{2} - 21 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 32 T^{2} + 546 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 40 T^{2} + 834 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 28 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 62 T^{2} + 1995 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 10 T^{2} + 1299 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 8 T + 72 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 20 T^{2} - 1146 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 8 T + 84 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 152 T^{2} + 9906 T^{4} + 152 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 158 T^{2} + 11211 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 38 T^{2} + 6171 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 208 T^{2} + 17970 T^{4} - 208 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 10 T + p T^{2} )^{8} \) |
| 71 | \( ( 1 - 176 T^{2} + 15234 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 220 T^{2} + 21606 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 14 T + 189 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 278 T^{2} + 32811 T^{4} + 278 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 190 T^{2} + 20643 T^{4} - 190 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.39786918906319396913296434696, −3.10284247477870233367588560940, −3.03357264331842822635424749992, −2.85676505142408061756831212380, −2.76737952802638415738256398601, −2.68572115816442472661767135595, −2.50185181514094788940795027570, −2.48211103233582094895701255927, −2.19171934430473069091317529684, −2.14603077676094581677458177926, −2.11760118990055219289141480650, −1.97452190343926226514562344643, −1.88397087121590489365828848818, −1.82001496355857464923156134461, −1.80035373753227269061797918010, −1.78488839669787893384490655466, −1.24320155624025672833199812804, −0.988785527597009818503995775931, −0.873636422810191570444985557342, −0.870513774538379316294797767107, −0.838318922216075097983801542521, −0.78848323948171955895583714197, −0.70958254669632545335344746530, −0.29712655509578796472748595100, −0.16502663567503521367594274302,
0.16502663567503521367594274302, 0.29712655509578796472748595100, 0.70958254669632545335344746530, 0.78848323948171955895583714197, 0.838318922216075097983801542521, 0.870513774538379316294797767107, 0.873636422810191570444985557342, 0.988785527597009818503995775931, 1.24320155624025672833199812804, 1.78488839669787893384490655466, 1.80035373753227269061797918010, 1.82001496355857464923156134461, 1.88397087121590489365828848818, 1.97452190343926226514562344643, 2.11760118990055219289141480650, 2.14603077676094581677458177926, 2.19171934430473069091317529684, 2.48211103233582094895701255927, 2.50185181514094788940795027570, 2.68572115816442472661767135595, 2.76737952802638415738256398601, 2.85676505142408061756831212380, 3.03357264331842822635424749992, 3.10284247477870233367588560940, 3.39786918906319396913296434696
Plot not available for L-functions of degree greater than 10.
