| L(s) = 1 | − 16·4-s − 48·5-s − 768·16-s + 768·20-s − 780·23-s − 4.52e3·25-s − 4.78e3·31-s + 1.02e4·37-s + 2.65e4·47-s − 3.36e4·49-s + 468·53-s − 7.62e4·59-s + 2.86e4·64-s + 5.86e4·67-s + 1.46e5·71-s + 3.68e4·80-s + 1.97e5·89-s + 1.24e4·92-s + 3.69e4·97-s + 7.23e4·100-s − 3.65e5·103-s + 2.71e5·113-s + 3.74e4·115-s + 7.65e4·124-s + 3.94e5·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.858·5-s − 3/4·16-s + 0.429·20-s − 0.307·23-s − 1.44·25-s − 0.894·31-s + 1.23·37-s + 1.75·47-s − 1.99·49-s + 0.0228·53-s − 2.85·59-s + 7/8·64-s + 1.59·67-s + 3.44·71-s + 0.643·80-s + 2.64·89-s + 0.153·92-s + 0.398·97-s + 0.723·100-s − 3.39·103-s + 2.00·113-s + 0.263·115-s + 0.447·124-s + 2.25·125-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.454980788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.454980788\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + p^{4} T^{2} + p^{10} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 24 T + p^{5} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 33611 T^{2} + p^{10} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 693434 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 995746 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2870531 T^{2} + p^{10} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 390 T + p^{5} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 29847598 T^{2} + p^{10} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2393 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 5137 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 39151610 T^{2} + p^{10} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 258722186 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 13266 T + p^{5} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 234 T + p^{5} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 38118 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 888695927 T^{2} + p^{10} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 29335 T + p^{5} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 73212 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 885792889 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 5723623475 T^{2} + p^{10} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 2133793594 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 98892 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18485 T + p^{5} T^{2} )^{2} \) |
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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
−9.288148496912642379987772711803, −9.108158683646274652296236379399, −8.360253885933892570018891753830, −8.095063676176934611794960884151, −7.78463505902306257381989480976, −7.42135597894278142608593935549, −6.87299267577281773681775181525, −6.37825919912638551180910468099, −6.03354848976830769143364061514, −5.49265731900087877218354349246, −4.90674090588837992531729743082, −4.63666492074768476484386415512, −3.99435200541411475103637511290, −3.77910235926890982400286816442, −3.31260767932327411142548676759, −2.52241062305538025494418194426, −2.08958706739358207966508303092, −1.52089283234373654809076934150, −0.60783696006015106455472924303, −0.36650662029011522591730654237,
0.36650662029011522591730654237, 0.60783696006015106455472924303, 1.52089283234373654809076934150, 2.08958706739358207966508303092, 2.52241062305538025494418194426, 3.31260767932327411142548676759, 3.77910235926890982400286816442, 3.99435200541411475103637511290, 4.63666492074768476484386415512, 4.90674090588837992531729743082, 5.49265731900087877218354349246, 6.03354848976830769143364061514, 6.37825919912638551180910468099, 6.87299267577281773681775181525, 7.42135597894278142608593935549, 7.78463505902306257381989480976, 8.095063676176934611794960884151, 8.360253885933892570018891753830, 9.108158683646274652296236379399, 9.288148496912642379987772711803
