| L(s) = 1 | − 2·2-s − 4·5-s − 6·7-s + 4·8-s + 9-s + 8·10-s − 8·11-s − 2·13-s + 12·14-s − 4·16-s + 2·17-s − 2·18-s − 12·19-s + 16·22-s + 12·23-s + 3·25-s + 4·26-s + 2·31-s − 4·34-s + 24·35-s + 2·37-s + 24·38-s − 16·40-s − 4·41-s − 6·43-s − 4·45-s − 24·46-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.78·5-s − 2.26·7-s + 1.41·8-s + 1/3·9-s + 2.52·10-s − 2.41·11-s − 0.554·13-s + 3.20·14-s − 16-s + 0.485·17-s − 0.471·18-s − 2.75·19-s + 3.41·22-s + 2.50·23-s + 3/5·25-s + 0.784·26-s + 0.359·31-s − 0.685·34-s + 4.05·35-s + 0.328·37-s + 3.89·38-s − 2.52·40-s − 0.624·41-s − 0.914·43-s − 0.596·45-s − 3.53·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19881 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19881 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) |
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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
−16.3084950271, −15.7159115639, −15.5159935893, −15.1680730202, −14.6489059291, −13.3552681657, −13.2982222373, −12.7859799900, −12.6657932749, −11.9075398839, −11.0645501220, −10.5981068261, −10.1950027975, −9.89655043622, −9.22229106146, −8.53725475056, −8.40526764581, −7.63249491007, −7.30736849694, −6.71008245044, −5.83486052932, −4.75349830420, −4.36466935125, −3.31847810115, −2.77908676833, 0, 0,
2.77908676833, 3.31847810115, 4.36466935125, 4.75349830420, 5.83486052932, 6.71008245044, 7.30736849694, 7.63249491007, 8.40526764581, 8.53725475056, 9.22229106146, 9.89655043622, 10.1950027975, 10.5981068261, 11.0645501220, 11.9075398839, 12.6657932749, 12.7859799900, 13.2982222373, 13.3552681657, 14.6489059291, 15.1680730202, 15.5159935893, 15.7159115639, 16.3084950271
