| L(s) = 1 | + 2·2-s + 2·4-s − 5-s + 2·7-s + 4·8-s − 2·10-s − 11-s − 4·13-s + 4·14-s + 8·16-s − 4·17-s − 2·20-s − 2·22-s + 23-s + 5·25-s − 8·26-s + 4·28-s − 7·31-s + 8·32-s − 8·34-s − 2·35-s + 6·37-s − 4·40-s + 8·41-s + 6·43-s − 2·44-s + 2·46-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.447·5-s + 0.755·7-s + 1.41·8-s − 0.632·10-s − 0.301·11-s − 1.10·13-s + 1.06·14-s + 2·16-s − 0.970·17-s − 0.447·20-s − 0.426·22-s + 0.208·23-s + 25-s − 1.56·26-s + 0.755·28-s − 1.25·31-s + 1.41·32-s − 1.37·34-s − 0.338·35-s + 0.986·37-s − 0.632·40-s + 1.24·41-s + 0.914·43-s − 0.301·44-s + 0.294·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 793881 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 793881 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.474700716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.474700716\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 6 T - 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 12 T + 83 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 7 T - 48 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
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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
−10.66295659293859806497800795290, −10.08743329892183934504542722219, −9.365701588977284945576055223209, −9.200389980300682870852647434820, −8.615933931568244902023499452891, −7.914510205276178384785258175379, −7.67774524044085329807655639786, −7.44131179812502982966235398717, −6.96287025037577745235969306248, −6.25024784498824974826541263962, −5.98104958018836848049172146501, −5.25704252722515264182910305830, −4.80420337545370249784913092303, −4.61347898967510088240709179001, −4.36835539561987769244284227735, −3.45688696280049179578436652359, −3.23120566598701975081249975262, −2.19989841012409930974517391371, −2.01485887818423098635907290802, −0.821839682517520122027716419683,
0.821839682517520122027716419683, 2.01485887818423098635907290802, 2.19989841012409930974517391371, 3.23120566598701975081249975262, 3.45688696280049179578436652359, 4.36835539561987769244284227735, 4.61347898967510088240709179001, 4.80420337545370249784913092303, 5.25704252722515264182910305830, 5.98104958018836848049172146501, 6.25024784498824974826541263962, 6.96287025037577745235969306248, 7.44131179812502982966235398717, 7.67774524044085329807655639786, 7.914510205276178384785258175379, 8.615933931568244902023499452891, 9.200389980300682870852647434820, 9.365701588977284945576055223209, 10.08743329892183934504542722219, 10.66295659293859806497800795290
