| L(s) = 1 | − 40·25-s + 2.06e3·49-s + 3.12e3·73-s − 3.41e3·97-s + 9.68e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.55e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | − 0.319·25-s + 6.02·49-s + 5.01·73-s − 3.57·97-s + 7.27·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 7.07·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 0.000292·227-s + 0.000288·229-s + 0.000281·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(60.74302225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(60.74302225\) |
| \(L(\frac{5}{2})\) |
|
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 | \( ( 1 + 2 p T^{2} + p^{6} T^{4} )^{4} \) |
| 7 | \( ( 1 - 517 T^{2} + p^{6} T^{4} )^{4} \) |
| 11 | \( ( 1 - 2422 T^{2} + p^{6} T^{4} )^{4} \) |
| 13 | \( ( 1 - 7 p T + p^{3} T^{2} )^{4}( 1 + 7 p T + p^{3} T^{2} )^{4} \) |
| 17 | \( ( 1 - 9106 T^{2} + p^{6} T^{4} )^{4} \) |
| 19 | \( ( 1 + 7091 T^{2} + p^{6} T^{4} )^{4} \) |
| 23 | \( ( 1 + 23614 T^{2} + p^{6} T^{4} )^{4} \) |
| 29 | \( ( 1 + 14218 T^{2} + p^{6} T^{4} )^{4} \) |
| 31 | \( ( 1 - 29998 T^{2} + p^{6} T^{4} )^{4} \) |
| 37 | \( ( 1 - 100223 T^{2} + p^{6} T^{4} )^{4} \) |
| 41 | \( ( 1 - 65842 T^{2} + p^{6} T^{4} )^{4} \) |
| 43 | \( ( 1 + 120026 T^{2} + p^{6} T^{4} )^{4} \) |
| 47 | \( ( 1 + 120526 T^{2} + p^{6} T^{4} )^{4} \) |
| 53 | \( ( 1 + 20314 T^{2} + p^{6} T^{4} )^{4} \) |
| 59 | \( ( 1 + 75242 T^{2} + p^{6} T^{4} )^{4} \) |
| 61 | \( ( 1 - 233639 T^{2} + p^{6} T^{4} )^{4} \) |
| 67 | \( ( 1 + 601163 T^{2} + p^{6} T^{4} )^{4} \) |
| 71 | \( ( 1 + 367342 T^{2} + p^{6} T^{4} )^{4} \) |
| 73 | \( ( 1 - 391 T + p^{3} T^{2} )^{8} \) |
| 79 | \( ( 1 - 815509 T^{2} + p^{6} T^{4} )^{4} \) |
| 83 | \( ( 1 - 1128214 T^{2} + p^{6} T^{4} )^{4} \) |
| 89 | \( ( 1 + 318782 T^{2} + p^{6} T^{4} )^{4} \) |
| 97 | \( ( 1 + 427 T + p^{3} T^{2} )^{8} \) |
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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
−3.60384016923051637599465042147, −3.33651767375785512328784329034, −3.16327302930514726005893983566, −3.10915235115189861687543476966, −3.03968027926798216046601839810, −2.88697848130741694990970970152, −2.76294498516896267703839154685, −2.72333871833256273564462987436, −2.56779062215656108152555684190, −2.20690998873861301001045797870, −2.20573528833785962101900327331, −2.12870468817607907956090860398, −1.98957016208659169448268239971, −1.82199994223844591826110100803, −1.75930977912992682704015611376, −1.58879736915747624994580697505, −1.43848341715490876283260790035, −1.20370022153543332330316575264, −0.857344083005708746432610614493, −0.819491260106450610304736498798, −0.72618684038995744833981885771, −0.62861314590456621352768003311, −0.45489948575800844242700943665, −0.40994099350387857002221462302, −0.33533880034637176823788594538,
0.33533880034637176823788594538, 0.40994099350387857002221462302, 0.45489948575800844242700943665, 0.62861314590456621352768003311, 0.72618684038995744833981885771, 0.819491260106450610304736498798, 0.857344083005708746432610614493, 1.20370022153543332330316575264, 1.43848341715490876283260790035, 1.58879736915747624994580697505, 1.75930977912992682704015611376, 1.82199994223844591826110100803, 1.98957016208659169448268239971, 2.12870468817607907956090860398, 2.20573528833785962101900327331, 2.20690998873861301001045797870, 2.56779062215656108152555684190, 2.72333871833256273564462987436, 2.76294498516896267703839154685, 2.88697848130741694990970970152, 3.03968027926798216046601839810, 3.10915235115189861687543476966, 3.16327302930514726005893983566, 3.33651767375785512328784329034, 3.60384016923051637599465042147
Plot not available for L-functions of degree greater than 10.
