| L(s) = 1 | + 3·4-s − 164·13-s − 55·16-s + 40·19-s − 174·25-s − 312·31-s + 372·37-s + 328·43-s − 492·52-s − 1.58e3·61-s − 357·64-s − 88·67-s − 252·73-s + 120·76-s − 1.42e3·79-s − 1.59e3·97-s − 522·100-s + 1.83e3·103-s − 684·109-s − 762·121-s − 936·124-s + 127-s + 131-s + 137-s + 139-s + 1.11e3·148-s + 149-s + ⋯ |
| L(s) = 1 | + 3/8·4-s − 3.49·13-s − 0.859·16-s + 0.482·19-s − 1.39·25-s − 1.80·31-s + 1.65·37-s + 1.16·43-s − 1.31·52-s − 3.31·61-s − 0.697·64-s − 0.160·67-s − 0.404·73-s + 0.181·76-s − 2.02·79-s − 1.67·97-s − 0.521·100-s + 1.75·103-s − 0.601·109-s − 0.572·121-s − 0.677·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.619·148-s + 0.000549·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 174 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 762 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 82 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 3670 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 7234 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10806 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 156 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 186 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 110406 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 164 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 13970 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 273130 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 386134 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 790 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 44 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 518146 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 126 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 712 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1001450 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 709626 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 798 T + p^{3} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.46559372525091224731249065020, −9.899663658060037205397210913282, −9.573169880702170772963716470530, −9.322126065354923390053054972524, −8.851801670679080149660290467972, −7.909134751528004526970981458762, −7.49819997417493070043462556601, −7.41647412442285102279247035085, −6.98883468530533569084981380967, −6.16015286910046052516033990741, −5.73427954776251340983509889552, −5.16975761149035710415937994111, −4.49473345724366268152022215927, −4.38070186325760608572410437539, −3.32449659341915639420278778702, −2.49829392819491818540720478451, −2.40500208686536614592774595713, −1.52048386907106017498380910051, 0, 0,
1.52048386907106017498380910051, 2.40500208686536614592774595713, 2.49829392819491818540720478451, 3.32449659341915639420278778702, 4.38070186325760608572410437539, 4.49473345724366268152022215927, 5.16975761149035710415937994111, 5.73427954776251340983509889552, 6.16015286910046052516033990741, 6.98883468530533569084981380967, 7.41647412442285102279247035085, 7.49819997417493070043462556601, 7.909134751528004526970981458762, 8.851801670679080149660290467972, 9.322126065354923390053054972524, 9.573169880702170772963716470530, 9.899663658060037205397210913282, 10.46559372525091224731249065020
