| L(s) = 1 | + 72·7-s − 9·9-s − 44·17-s + 288·23-s + 54·25-s − 216·31-s + 892·41-s + 144·47-s + 3.20e3·49-s − 648·63-s − 1.15e3·71-s + 108·73-s − 1.94e3·79-s + 81·81-s − 692·89-s − 2.26e3·97-s − 2.66e3·103-s − 892·113-s − 3.16e3·119-s + 1.36e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 396·153-s + ⋯ |
| L(s) = 1 | + 3.88·7-s − 1/3·9-s − 0.627·17-s + 2.61·23-s + 0.431·25-s − 1.25·31-s + 3.39·41-s + 0.446·47-s + 9.33·49-s − 1.29·63-s − 1.92·71-s + 0.173·73-s − 2.76·79-s + 1/9·81-s − 0.824·89-s − 2.37·97-s − 2.54·103-s − 0.742·113-s − 2.44·119-s + 1.02·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.209·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.464094243\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.464094243\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 54 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 36 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 1366 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 1478 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 22 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 12422 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 144 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 46278 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 108 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 396 T + p^{3} T^{2} )( 1 + 396 T + p^{3} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 446 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 95510 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 72 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 297270 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 57098 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 236806 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 569126 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 576 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 54 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 972 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 675718 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 346 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 1134 T + p^{3} T^{2} )^{2} \) |
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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
−10.54320161844613419402047713464, −9.380358695295879488122970587467, −9.301748530857344900471331715087, −8.645648820263248197326077566818, −8.435943065398367983412681695590, −8.179213538535867421005901245481, −7.43420742110695606596924939192, −7.24874021355817844151069268294, −7.06900518628010736918103792410, −5.77120923822547037439717060733, −5.70108042280637479190940808430, −5.16181901221327851003918733322, −4.68607437365297943096769060220, −4.38143662349385348078188108666, −4.03451248207390447857124496465, −2.76449935303120174932572390572, −2.55468015506472130513590467092, −1.63688017792698419700378782910, −1.36276131890476770444234617091, −0.75375405850339350467830960990,
0.75375405850339350467830960990, 1.36276131890476770444234617091, 1.63688017792698419700378782910, 2.55468015506472130513590467092, 2.76449935303120174932572390572, 4.03451248207390447857124496465, 4.38143662349385348078188108666, 4.68607437365297943096769060220, 5.16181901221327851003918733322, 5.70108042280637479190940808430, 5.77120923822547037439717060733, 7.06900518628010736918103792410, 7.24874021355817844151069268294, 7.43420742110695606596924939192, 8.179213538535867421005901245481, 8.435943065398367983412681695590, 8.645648820263248197326077566818, 9.301748530857344900471331715087, 9.380358695295879488122970587467, 10.54320161844613419402047713464
