| L(s) = 1 | − 4-s + 5-s − 2·9-s + 2·19-s − 20-s − 2·31-s + 2·36-s − 2·41-s − 2·45-s − 49-s − 2·59-s + 64-s + 2·71-s − 2·76-s + 3·81-s + 2·95-s + 2·101-s + 2·109-s + 2·121-s + 2·124-s − 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 4-s + 5-s − 2·9-s + 2·19-s − 20-s − 2·31-s + 2·36-s − 2·41-s − 2·45-s − 49-s − 2·59-s + 64-s + 2·71-s − 2·76-s + 3·81-s + 2·95-s + 2·101-s + 2·109-s + 2·121-s + 2·124-s − 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3694422763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3694422763\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−14.01728269487878139040519980875, −13.24129811164260988978612131150, −12.46543236559967768776943439337, −11.95316490223993661237857601247, −11.32697481286746022943537966114, −11.15479282252388540651046991477, −10.30377698260116748695414024712, −9.749128852540363752407603347420, −9.234470675715962152120679315755, −9.069309539962566150968174134441, −8.406564653898943386903774874381, −7.87127156377874814137322110235, −7.16240109251553910414852154377, −6.28352037519677019263430211854, −5.79099571878796810866265396472, −5.14454408149836053986851032894, −4.99559637066076860419486785128, −3.57720643177974959481399999539, −3.14553519709855962013572144934, −1.98470118185236666196526716097,
1.98470118185236666196526716097, 3.14553519709855962013572144934, 3.57720643177974959481399999539, 4.99559637066076860419486785128, 5.14454408149836053986851032894, 5.79099571878796810866265396472, 6.28352037519677019263430211854, 7.16240109251553910414852154377, 7.87127156377874814137322110235, 8.406564653898943386903774874381, 9.069309539962566150968174134441, 9.234470675715962152120679315755, 9.749128852540363752407603347420, 10.30377698260116748695414024712, 11.15479282252388540651046991477, 11.32697481286746022943537966114, 11.95316490223993661237857601247, 12.46543236559967768776943439337, 13.24129811164260988978612131150, 14.01728269487878139040519980875
