L(s) = 1 | − 2-s + 4-s − 2·5-s − 8-s + 9-s + 2·10-s − 2·13-s + 16-s − 12·17-s − 18-s − 2·20-s + 3·25-s + 2·26-s − 20·29-s − 32-s + 12·34-s + 36-s − 12·37-s + 2·40-s + 4·41-s − 2·45-s − 14·49-s − 3·50-s − 2·52-s − 12·53-s + 20·58-s + 12·61-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.554·13-s + 1/4·16-s − 2.91·17-s − 0.235·18-s − 0.447·20-s + 3/5·25-s + 0.392·26-s − 3.71·29-s − 0.176·32-s + 2.05·34-s + 1/6·36-s − 1.97·37-s + 0.316·40-s + 0.624·41-s − 0.298·45-s − 2·49-s − 0.424·50-s − 0.277·52-s − 1.64·53-s + 2.62·58-s + 1.53·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−7.48651432603210725392086041861, −7.13489552716111364035097044906, −7.05881776326155092462693764661, −6.27139115246731968799115000690, −6.13130217685388014086757536825, −5.09225276175166925177215351932, −5.02996251545709838502394177698, −4.33322549174996816277799124115, −3.75949148792159340735853491565, −3.53361530480953533803089286496, −2.61347722092406101275253466214, −2.01716871449082204952154497689, −1.63593081225043651097784490840, 0, 0,
1.63593081225043651097784490840, 2.01716871449082204952154497689, 2.61347722092406101275253466214, 3.53361530480953533803089286496, 3.75949148792159340735853491565, 4.33322549174996816277799124115, 5.02996251545709838502394177698, 5.09225276175166925177215351932, 6.13130217685388014086757536825, 6.27139115246731968799115000690, 7.05881776326155092462693764661, 7.13489552716111364035097044906, 7.48651432603210725392086041861