| L(s) = 1 | − 8·7-s − 2·16-s + 8·31-s + 4·49-s − 16·73-s − 16·79-s − 8·97-s − 16·103-s + 16·112-s + 24·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 3.02·7-s − 1/2·16-s + 1.43·31-s + 4/7·49-s − 1.87·73-s − 1.80·79-s − 0.812·97-s − 1.57·103-s + 1.51·112-s + 2.18·121-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 + 3.38·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.113272747\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.113272747\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T^{4} + p^{4} T^{8} \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 + 2 T + 9 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 12 T^{2} + 62 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 11 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 20 T^{2} + 462 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 26 T^{2} + 291 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 60 T^{2} + 1742 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 68 T^{2} + 2622 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 2 T + 57 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 84 T^{2} + 4526 T^{4} + 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 122 T^{2} + 6819 T^{4} - 122 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 60 T^{2} + 3782 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 132 T^{2} + 9374 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 108 T^{2} + 8342 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 134 T^{2} + 9531 T^{4} - 134 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 58 T^{2} + 4419 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 36 T^{2} + 9806 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 140 T^{2} + 15222 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 36 T^{2} + 6566 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 2 T - 21 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−3.78673269623350533239327895471, −3.78083086793718529230094171329, −3.77559467941451907212139011409, −3.50610653305989974831247170487, −3.19945879292229262876923508528, −3.12713704020129484315146803348, −3.10300854354423157112961738556, −3.06872271439367829944387205021, −3.04457038598820373208338532932, −2.88523934697075753531873328202, −2.67019098492418360290094941190, −2.63597182264489210143635697662, −2.50753049521749821033022489193, −2.15730334899564456653648152212, −2.06071159504008873876306112432, −1.82407933104550253481718032268, −1.73101791808793490699513164683, −1.62084915978282664017140808820, −1.61843655185879572596156433056, −1.19894069870310017063058508750, −1.05457960485009920290374347974, −0.69770601434804257200293022470, −0.53114621172528910950794458792, −0.37742151027453818120113416986, −0.24905098608523480154529344892,
0.24905098608523480154529344892, 0.37742151027453818120113416986, 0.53114621172528910950794458792, 0.69770601434804257200293022470, 1.05457960485009920290374347974, 1.19894069870310017063058508750, 1.61843655185879572596156433056, 1.62084915978282664017140808820, 1.73101791808793490699513164683, 1.82407933104550253481718032268, 2.06071159504008873876306112432, 2.15730334899564456653648152212, 2.50753049521749821033022489193, 2.63597182264489210143635697662, 2.67019098492418360290094941190, 2.88523934697075753531873328202, 3.04457038598820373208338532932, 3.06872271439367829944387205021, 3.10300854354423157112961738556, 3.12713704020129484315146803348, 3.19945879292229262876923508528, 3.50610653305989974831247170487, 3.77559467941451907212139011409, 3.78083086793718529230094171329, 3.78673269623350533239327895471
Plot not available for L-functions of degree greater than 10.
