| L(s) = 1 | − 2-s − 6·5-s − 7-s + 8-s + 6·10-s + 5·13-s + 14-s − 16-s + 6·17-s + 4·19-s + 3·23-s + 17·25-s − 5·26-s + 6·29-s − 20·31-s − 6·34-s + 6·35-s − 8·37-s − 4·38-s − 6·40-s − 8·43-s − 3·46-s − 12·47-s − 17·50-s − 24·53-s − 56-s − 6·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 2.68·5-s − 0.377·7-s + 0.353·8-s + 1.89·10-s + 1.38·13-s + 0.267·14-s − 1/4·16-s + 1.45·17-s + 0.917·19-s + 0.625·23-s + 17/5·25-s − 0.980·26-s + 1.11·29-s − 3.59·31-s − 1.02·34-s + 1.01·35-s − 1.31·37-s − 0.648·38-s − 0.948·40-s − 1.21·43-s − 0.442·46-s − 1.75·47-s − 2.40·50-s − 3.29·53-s − 0.133·56-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 3 T - 62 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.115774427588332292137399176638, −8.748276987373958787166041643777, −8.195365691898283356273906745321, −8.022530740941099095309154796721, −7.76925061340699859271400623055, −7.40330414381508375584473572646, −6.85286070345043910990162241056, −6.73095173934942509089957992803, −5.94706332934839104246854760940, −5.43680035632952664373421911823, −4.97777727641876856286263336221, −4.57391976956826796824376253699, −3.84423679172735358704651473898, −3.54733071740672976248642527689, −3.37587448310447171637300057402, −2.99602865912465226327249212230, −1.43660887997356190527905534323, −1.40320390684268550963506944719, 0, 0,
1.40320390684268550963506944719, 1.43660887997356190527905534323, 2.99602865912465226327249212230, 3.37587448310447171637300057402, 3.54733071740672976248642527689, 3.84423679172735358704651473898, 4.57391976956826796824376253699, 4.97777727641876856286263336221, 5.43680035632952664373421911823, 5.94706332934839104246854760940, 6.73095173934942509089957992803, 6.85286070345043910990162241056, 7.40330414381508375584473572646, 7.76925061340699859271400623055, 8.022530740941099095309154796721, 8.195365691898283356273906745321, 8.748276987373958787166041643777, 9.115774427588332292137399176638
