| L(s) = 1 | − 4·4-s − 4·9-s − 2·16-s + 16·36-s + 40·49-s + 24·64-s + 64·79-s + 16·81-s − 80·109-s + 64·121-s + 127-s + 131-s + 137-s + 139-s + 8·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 160·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 160·196-s + ⋯ |
| L(s) = 1 | − 2·4-s − 4/3·9-s − 1/2·16-s + 8/3·36-s + 40/7·49-s + 3·64-s + 7.20·79-s + 16/9·81-s − 7.66·109-s + 5.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 12.3·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 11.4·196-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04056453419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04056453419\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + 2 T^{2} - 2 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| good | 2 | \( ( 1 + T^{2} + 3 T^{4} + p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 11 | \( ( 1 - 16 T^{2} + 285 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 13 | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{8} \) |
| 17 | \( ( 1 - 26 T^{2} + 558 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 19 | \( ( 1 - 10 T^{2} + 558 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 23 | \( ( 1 + 64 T^{2} + 1893 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 29 | \( ( 1 - 88 T^{2} + 3429 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 31 | \( ( 1 - 58 T^{2} + 2574 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 37 | \( ( 1 - 2 p T^{2} + 2763 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} )^{4} \) |
| 41 | \( ( 1 + 38 T^{2} + 2022 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 43 | \( ( 1 - 122 T^{2} + 7083 T^{4} - 122 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 47 | \( ( 1 - 146 T^{2} + 9558 T^{4} - 146 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 53 | \( ( 1 + 16 T^{2} - 1122 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 59 | \( ( 1 + 68 T^{2} + 7362 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 61 | \( ( 1 - 178 T^{2} + 15174 T^{4} - 178 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 67 | \( ( 1 - 50 T^{2} + 1203 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 71 | \( ( 1 - 256 T^{2} + 26445 T^{4} - 256 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 73 | \( ( 1 - 8 T^{2} - 1422 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 79 | \( ( 1 - 8 T + 153 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{8} \) |
| 83 | \( ( 1 - 38 T^{2} + 4878 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 89 | \( ( 1 + 188 T^{2} + 23922 T^{4} + 188 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 97 | \( ( 1 + 250 T^{2} + 29718 T^{4} + 250 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.99571335218248148935653505666, −2.82224520617536495879128557205, −2.75243534002752259158826003568, −2.64021013752069214002282219398, −2.53806800813149114098450098161, −2.49929197559709090496465974826, −2.45883302131878145693247727974, −2.40135751821303900844923399628, −2.28860016229234199131964939212, −2.21827998914425266084202422274, −2.06184362086127818433135434626, −2.04930744365160506444409857172, −1.86684791248305600908314874923, −1.80635054252704377120056768335, −1.61643969508464119109666710840, −1.59223454252716779021796001491, −1.50703478872330246170754156174, −1.18907100932464992625011913349, −0.957227517005743187538836950826, −0.912775764673700050471878979403, −0.888731272681181487460422418569, −0.790496798338438195516719997049, −0.54401575788286510093822544853, −0.35665445380557668384592373830, −0.03024089265334751831248067701,
0.03024089265334751831248067701, 0.35665445380557668384592373830, 0.54401575788286510093822544853, 0.790496798338438195516719997049, 0.888731272681181487460422418569, 0.912775764673700050471878979403, 0.957227517005743187538836950826, 1.18907100932464992625011913349, 1.50703478872330246170754156174, 1.59223454252716779021796001491, 1.61643969508464119109666710840, 1.80635054252704377120056768335, 1.86684791248305600908314874923, 2.04930744365160506444409857172, 2.06184362086127818433135434626, 2.21827998914425266084202422274, 2.28860016229234199131964939212, 2.40135751821303900844923399628, 2.45883302131878145693247727974, 2.49929197559709090496465974826, 2.53806800813149114098450098161, 2.64021013752069214002282219398, 2.75243534002752259158826003568, 2.82224520617536495879128557205, 2.99571335218248148935653505666
Plot not available for L-functions of degree greater than 10.
