| L(s) = 1 | + 2·2-s + 3·5-s − 29·7-s − 8·8-s + 6·10-s − 57·11-s − 20·13-s − 58·14-s − 16·16-s + 144·17-s − 212·19-s − 114·22-s + 174·23-s + 125·25-s − 40·26-s − 210·29-s − 47·31-s + 288·34-s − 87·35-s + 4·37-s − 424·38-s − 24·40-s − 6·41-s − 218·43-s + 348·46-s + 474·47-s + 343·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.268·5-s − 1.56·7-s − 0.353·8-s + 0.189·10-s − 1.56·11-s − 0.426·13-s − 1.10·14-s − 1/4·16-s + 2.05·17-s − 2.55·19-s − 1.10·22-s + 1.57·23-s + 25-s − 0.301·26-s − 1.34·29-s − 0.272·31-s + 1.45·34-s − 0.420·35-s + 0.0177·37-s − 1.81·38-s − 0.0948·40-s − 0.0228·41-s − 0.773·43-s + 1.11·46-s + 1.47·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.350378059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.350378059\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T - 116 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 29 T + 498 T^{2} + 29 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 57 T + 1918 T^{2} + 57 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 20 T - 1797 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 72 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 106 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 174 T + 18109 T^{2} - 174 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 210 T + 19711 T^{2} + 210 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 47 T - 27582 T^{2} + 47 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 68885 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 218 T - 31983 T^{2} + 218 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 474 T + 120853 T^{2} - 474 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 81 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 84 T - 198323 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 56 T - 223845 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 142 T - 280599 T^{2} - 142 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 360 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 1159 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 160 T - 467439 T^{2} - 160 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 735 T - 31562 T^{2} - 735 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 954 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 191 T - 876192 T^{2} + 191 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−12.63794796033164151739029800276, −12.61296096149047539935016368030, −11.88974068758758808788440486702, −11.04194359008823535788244057662, −10.52783612084083439190210376705, −10.23620212604956414302443286535, −9.778862590215742392642352235689, −8.940738102577678852085307969464, −8.797161683412059930877580697832, −7.84733939103922196449183318787, −7.29351290716720015690662017400, −6.79783478891792523767912917272, −5.91678404381643879647298706325, −5.76563549019284187962409221990, −4.96677152354747967274466085168, −4.36165175241548490028176415598, −3.20443399351195949039685153374, −3.14468551065697517906697075772, −2.12169565440095327355698475103, −0.46718200580614804687676123182,
0.46718200580614804687676123182, 2.12169565440095327355698475103, 3.14468551065697517906697075772, 3.20443399351195949039685153374, 4.36165175241548490028176415598, 4.96677152354747967274466085168, 5.76563549019284187962409221990, 5.91678404381643879647298706325, 6.79783478891792523767912917272, 7.29351290716720015690662017400, 7.84733939103922196449183318787, 8.797161683412059930877580697832, 8.940738102577678852085307969464, 9.778862590215742392642352235689, 10.23620212604956414302443286535, 10.52783612084083439190210376705, 11.04194359008823535788244057662, 11.88974068758758808788440486702, 12.61296096149047539935016368030, 12.63794796033164151739029800276
