| L(s) = 1 | + 112·13-s + 496·25-s + 1.03e3·37-s + 1.34e3·49-s + 664·61-s − 384·73-s − 1.37e3·97-s − 5.04e3·109-s − 5.26e3·121-s + ⋯ |
| L(s) = 1 | + 2.38·13-s + 3.96·25-s + 4.58·37-s + 3.90·49-s + 1.39·61-s − 0.615·73-s − 1.44·97-s − 4.42·109-s − 3.95·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.407456837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.407456837\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 248 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 670 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 2630 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 28 T + p^{3} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 8368 T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 4502 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 14 p^{2} T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 30728 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 49582 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 258 T + p^{3} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 100304 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 63770 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 81154 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 267496 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 235526 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 166 T + p^{3} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 278902 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 43022 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 96 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 529102 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 470774 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 1355488 T^{2} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 344 T + p^{3} T^{2} )^{4} \) |
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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
−8.313478493112608114424093033133, −7.86613558685172141346686333996, −7.54942422737460129289662579579, −7.35006586197087875297750856576, −7.26025137585927782510088647429, −6.54683715235225082089502819409, −6.50515500754874280922172107004, −6.41503100948324274500357963754, −6.21137185882843658216575727244, −5.68584851426339273573568094542, −5.37471251149131807690540313011, −5.25642098293353848610405382203, −5.01312370229114751784732403016, −4.32041802136564856301935601895, −4.13018801301006468152084346080, −4.04468065944317345859453284236, −3.85103297384986685291833917011, −3.07618485808262106452055336416, −2.77970625308397851149727630405, −2.72773641143341153627425434420, −2.38263468386974758856587966038, −1.47835532575918849484727304337, −1.09192061804771925841204775965, −0.975212182961573734074513411095, −0.60099681679500047950769891034,
0.60099681679500047950769891034, 0.975212182961573734074513411095, 1.09192061804771925841204775965, 1.47835532575918849484727304337, 2.38263468386974758856587966038, 2.72773641143341153627425434420, 2.77970625308397851149727630405, 3.07618485808262106452055336416, 3.85103297384986685291833917011, 4.04468065944317345859453284236, 4.13018801301006468152084346080, 4.32041802136564856301935601895, 5.01312370229114751784732403016, 5.25642098293353848610405382203, 5.37471251149131807690540313011, 5.68584851426339273573568094542, 6.21137185882843658216575727244, 6.41503100948324274500357963754, 6.50515500754874280922172107004, 6.54683715235225082089502819409, 7.26025137585927782510088647429, 7.35006586197087875297750856576, 7.54942422737460129289662579579, 7.86613558685172141346686333996, 8.313478493112608114424093033133
