| L(s) = 1 | − 2-s + 3-s + 5-s − 6-s + 8-s − 10-s + 4·11-s + 4·13-s + 15-s − 16-s + 2·17-s + 4·19-s − 4·22-s + 8·23-s + 24-s − 4·26-s − 27-s − 4·29-s − 30-s + 4·33-s − 2·34-s − 6·37-s − 4·38-s + 4·39-s + 40-s + 12·41-s − 8·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.447·5-s − 0.408·6-s + 0.353·8-s − 0.316·10-s + 1.20·11-s + 1.10·13-s + 0.258·15-s − 1/4·16-s + 0.485·17-s + 0.917·19-s − 0.852·22-s + 1.66·23-s + 0.204·24-s − 0.784·26-s − 0.192·27-s − 0.742·29-s − 0.182·30-s + 0.696·33-s − 0.342·34-s − 0.986·37-s − 0.648·38-s + 0.640·39-s + 0.158·40-s + 1.87·41-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.864735287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.864735287\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10 T + 47 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−9.519887504079053762506374699101, −9.295750823661913670386081752524, −8.851679020421109373606291536455, −8.688940863120753977781666107085, −8.282251608742224887300198925180, −7.80346326554922398275290875419, −7.19879750485023496605020665094, −7.09018925116833993259616490301, −6.56864293583740053097223430119, −6.10148402515147787775145821729, −5.50804200017496366662888209901, −5.30500400971023015213175400539, −4.69895467960221506608691984382, −3.93332640130454631124970694484, −3.66923429137743312209543802025, −3.29731560102758626147248361476, −2.52751915285571896899033224520, −1.98312152670810907756760744300, −1.15417666880861415904274916006, −0.930525596402006859466527614738,
0.930525596402006859466527614738, 1.15417666880861415904274916006, 1.98312152670810907756760744300, 2.52751915285571896899033224520, 3.29731560102758626147248361476, 3.66923429137743312209543802025, 3.93332640130454631124970694484, 4.69895467960221506608691984382, 5.30500400971023015213175400539, 5.50804200017496366662888209901, 6.10148402515147787775145821729, 6.56864293583740053097223430119, 7.09018925116833993259616490301, 7.19879750485023496605020665094, 7.80346326554922398275290875419, 8.282251608742224887300198925180, 8.688940863120753977781666107085, 8.851679020421109373606291536455, 9.295750823661913670386081752524, 9.519887504079053762506374699101
