| L(s) = 1 | − 8·2-s − 18·3-s + 48·4-s + 98·5-s + 144·6-s + 98·7-s − 256·8-s + 243·9-s − 784·10-s + 242·11-s − 864·12-s + 182·13-s − 784·14-s − 1.76e3·15-s + 1.28e3·16-s − 1.24e3·17-s − 1.94e3·18-s − 66·19-s + 4.70e3·20-s − 1.76e3·21-s − 1.93e3·22-s − 4.18e3·23-s + 4.60e3·24-s + 953·25-s − 1.45e3·26-s − 2.91e3·27-s + 4.70e3·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.75·5-s + 1.63·6-s + 0.755·7-s − 1.41·8-s + 9-s − 2.47·10-s + 0.603·11-s − 1.73·12-s + 0.298·13-s − 1.06·14-s − 2.02·15-s + 5/4·16-s − 1.04·17-s − 1.41·18-s − 0.0419·19-s + 2.62·20-s − 0.872·21-s − 0.852·22-s − 1.64·23-s + 1.63·24-s + 0.304·25-s − 0.422·26-s − 0.769·27-s + 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 49 T + p^{5} T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 14 p T + 523043 T^{2} - 14 p^{6} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1244 T + 435754 T^{2} + 1244 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 66 T + 4896331 T^{2} + 66 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4184 T + 16736546 T^{2} + 4184 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 266 p T + 1141943 p T^{2} + 266 p^{6} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2476 T + 10890950 T^{2} + 2476 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 9158 T + 77410691 T^{2} + 9158 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 11208 T + 261066802 T^{2} - 11208 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 16904 T + 228701834 T^{2} + 16904 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 9814 T + 195653467 T^{2} - 9814 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 19664 T + 153388526 T^{2} - 19664 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2718 T + 1418880379 T^{2} - 2718 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 10556 T + 1714999470 T^{2} - 10556 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 47306 T + 2886026307 T^{2} - 47306 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 38904 T + 3984788590 T^{2} + 38904 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 393 T + p^{5} T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 90296 T + 5584097726 T^{2} - 90296 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 15220 T + 6637339630 T^{2} + 15220 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 79852 T + 12280128790 T^{2} + 79852 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 51856 T + 16296314598 T^{2} + 51856 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−9.896613637264770662874788687475, −9.687242226171776142922684206133, −9.229989948520659684262276867513, −8.901764702239552487769225353070, −8.207205309399675310474576026657, −7.87439096757196726595007378492, −7.15859462892231213203789074174, −6.78622405476033670949427454312, −6.28652481310466298040982546658, −5.93736363023709196317675354026, −5.40438182855057061684965094980, −5.22273063721057614516464230928, −3.96692552480239649427049412141, −3.91745263845277533953034959906, −2.28089489114242733862180653683, −2.25152310186483862685696167701, −1.39471995204358471140857777204, −1.37586777990798664265704625611, 0, 0,
1.37586777990798664265704625611, 1.39471995204358471140857777204, 2.25152310186483862685696167701, 2.28089489114242733862180653683, 3.91745263845277533953034959906, 3.96692552480239649427049412141, 5.22273063721057614516464230928, 5.40438182855057061684965094980, 5.93736363023709196317675354026, 6.28652481310466298040982546658, 6.78622405476033670949427454312, 7.15859462892231213203789074174, 7.87439096757196726595007378492, 8.207205309399675310474576026657, 8.901764702239552487769225353070, 9.229989948520659684262276867513, 9.687242226171776142922684206133, 9.896613637264770662874788687475
