| L(s) = 1 | − 192·29-s − 132·41-s − 76·49-s + 152·61-s + 348·89-s − 168·101-s − 232·109-s − 266·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 124·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 6.62·29-s − 3.21·41-s − 1.55·49-s + 2.49·61-s + 3.91·89-s − 1.66·101-s − 2.12·109-s − 2.19·121-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 − 0.733·169-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 + 0.00473·211-s + 0.00448·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4275727639\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4275727639\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 133 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 353 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 347 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 998 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 48 T + p^{2} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 422 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 2638 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 33 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 4178 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 4718 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 962 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 38 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 5603 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 4082 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 10633 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 10982 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 13643 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 87 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 6718 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−5.84804330902684179259795153947, −5.76107887673538031102690844878, −5.26333971296842693138756126757, −5.19739747807321049841641921236, −5.12760282544071499420075895209, −5.03198773370963465797802880722, −4.97619058762388406774871034051, −4.16414892362617334082086991324, −4.15969133975573235623672150517, −4.06282188571492697310761254841, −3.82604347112061374582143818538, −3.54725905507446013770265033359, −3.48343562634668943850074945106, −3.20552990288609383747813192327, −3.14171715649792707554173434975, −2.59285711250610552573598254232, −2.21617725096019443492468321422, −2.19196618811713529431789100122, −2.07625990917042871318891262017, −1.59187374630114864147028670870, −1.45978654832922880984408890961, −1.37575276815167406507520162540, −0.78415986165469525859305389956, −0.24091149775541908253729068384, −0.15078857822702685634839419929,
0.15078857822702685634839419929, 0.24091149775541908253729068384, 0.78415986165469525859305389956, 1.37575276815167406507520162540, 1.45978654832922880984408890961, 1.59187374630114864147028670870, 2.07625990917042871318891262017, 2.19196618811713529431789100122, 2.21617725096019443492468321422, 2.59285711250610552573598254232, 3.14171715649792707554173434975, 3.20552990288609383747813192327, 3.48343562634668943850074945106, 3.54725905507446013770265033359, 3.82604347112061374582143818538, 4.06282188571492697310761254841, 4.15969133975573235623672150517, 4.16414892362617334082086991324, 4.97619058762388406774871034051, 5.03198773370963465797802880722, 5.12760282544071499420075895209, 5.19739747807321049841641921236, 5.26333971296842693138756126757, 5.76107887673538031102690844878, 5.84804330902684179259795153947
