| L(s) = 1 | + 8·2-s + 48·4-s + 256·8-s − 162·9-s + 476·13-s + 1.28e3·16-s + 322·17-s − 1.29e3·18-s − 1.05e3·25-s + 3.80e3·26-s + 6.14e3·32-s + 2.57e3·34-s − 7.77e3·36-s − 4.80e3·49-s − 8.43e3·50-s + 2.28e4·52-s + 4.96e3·53-s + 2.86e4·64-s + 1.54e4·68-s − 4.14e4·72-s + 1.96e4·81-s + 1.95e4·89-s − 3.84e4·98-s − 5.05e4·100-s − 3.76e4·101-s + 1.21e5·104-s + 3.97e4·106-s + ⋯ |
| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s − 2·9-s + 2.81·13-s + 5·16-s + 1.11·17-s − 4·18-s − 1.68·25-s + 5.63·26-s + 6·32-s + 2.22·34-s − 6·36-s − 2·49-s − 3.37·50-s + 8.44·52-s + 1.76·53-s + 7·64-s + 3.34·68-s − 8·72-s + 3·81-s + 2.46·89-s − 4·98-s − 5.05·100-s − 3.68·101-s + 11.2·104-s + 3.53·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(8.370590151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.370590151\) |
| \(L(3)\) |
|
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 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 17 | $C_2$ | \( 1 - 322 T + p^{4} T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 14 T + p^{4} T^{2} )( 1 + 14 T + p^{4} T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 238 T + p^{4} T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 82 T + p^{4} T^{2} )( 1 + 82 T + p^{4} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2162 T + p^{4} T^{2} )( 1 + 2162 T + p^{4} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 3038 T + p^{4} T^{2} )( 1 + 3038 T + p^{4} T^{2} ) \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2482 T + p^{4} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6958 T + p^{4} T^{2} )( 1 + 6958 T + p^{4} T^{2} ) \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 1442 T + p^{4} T^{2} )( 1 + 1442 T + p^{4} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 9758 T + p^{4} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1918 T + p^{4} T^{2} )( 1 + 1918 T + p^{4} T^{2} ) \) |
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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
−14.09725799156025039896554999889, −13.62664179311966233811760373530, −13.57054305946541045716082698078, −12.82468319725972921348908385461, −11.99766836302080490572235653841, −11.67269786572505898892132185887, −11.20965950305944583470957878811, −10.83345788667365226103134540021, −10.10400523319896409971429110995, −8.976730426151440069471237777146, −8.086588472642463879346380079907, −7.943474600059238840260582772692, −6.64755815047291831657132538536, −6.01880620469936652318173193286, −5.80241719915847087344624279035, −5.12506620970902919003547153817, −3.70172862168894081795868666765, −3.63441121365638958749105094802, −2.61210054952995354294122515321, −1.35849107346732809115819097886,
1.35849107346732809115819097886, 2.61210054952995354294122515321, 3.63441121365638958749105094802, 3.70172862168894081795868666765, 5.12506620970902919003547153817, 5.80241719915847087344624279035, 6.01880620469936652318173193286, 6.64755815047291831657132538536, 7.943474600059238840260582772692, 8.086588472642463879346380079907, 8.976730426151440069471237777146, 10.10400523319896409971429110995, 10.83345788667365226103134540021, 11.20965950305944583470957878811, 11.67269786572505898892132185887, 11.99766836302080490572235653841, 12.82468319725972921348908385461, 13.57054305946541045716082698078, 13.62664179311966233811760373530, 14.09725799156025039896554999889
