| L(s) = 1 | + 14·9-s − 92·19-s − 132·31-s + 194·49-s + 156·61-s + 280·79-s + 115·81-s + 580·109-s + 340·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 226·169-s − 1.28e3·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 14/9·9-s − 4.84·19-s − 4.25·31-s + 3.95·49-s + 2.55·61-s + 3.54·79-s + 1.41·81-s + 5.32·109-s + 2.80·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.33·169-s − 7.53·171-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.873357068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.873357068\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 - 14 T^{2} + p^{4} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 - 97 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 170 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 113 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 186 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 23 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 1050 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1034 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 33 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 1618 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 2010 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 3649 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 2370 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 4266 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 3406 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 39 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 3791 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 9434 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 7294 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 9550 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 7650 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 18817 T^{2} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.88073119118536994344719488889, −6.75482483267630954374076762635, −6.15698734276339054596388130120, −6.12160939601965118047795796406, −6.07201868835543187258337993982, −5.54608549277465975191815446576, −5.43978089865414802344760698393, −5.39135108581129731622785555574, −4.82643302652638781520783789945, −4.61021291690372075104315352460, −4.34125641719378691634402817525, −4.34100468072992717919547037759, −3.97892338604426368140308358315, −3.80176572498985898740282173210, −3.60819347115328228803540934352, −3.42496234790667354451918092332, −2.92127175643012196596575310802, −2.45320062288465307626537305034, −2.03871160626783299013904520453, −2.00569157241661335682609786435, −1.89640581715012485815352247052, −1.84059687844131576518589366642, −0.78792819554787384063924859015, −0.61444982009574385522822968934, −0.42846827802913743869167884567,
0.42846827802913743869167884567, 0.61444982009574385522822968934, 0.78792819554787384063924859015, 1.84059687844131576518589366642, 1.89640581715012485815352247052, 2.00569157241661335682609786435, 2.03871160626783299013904520453, 2.45320062288465307626537305034, 2.92127175643012196596575310802, 3.42496234790667354451918092332, 3.60819347115328228803540934352, 3.80176572498985898740282173210, 3.97892338604426368140308358315, 4.34100468072992717919547037759, 4.34125641719378691634402817525, 4.61021291690372075104315352460, 4.82643302652638781520783789945, 5.39135108581129731622785555574, 5.43978089865414802344760698393, 5.54608549277465975191815446576, 6.07201868835543187258337993982, 6.12160939601965118047795796406, 6.15698734276339054596388130120, 6.75482483267630954374076762635, 6.88073119118536994344719488889
