| L(s) = 1 | − 2·2-s + 2·3-s − 5-s − 4·6-s − 6·7-s + 4·8-s − 3·9-s + 2·10-s + 6·11-s − 6·13-s + 12·14-s − 2·15-s − 4·16-s + 2·17-s + 6·18-s − 2·19-s − 12·21-s − 12·22-s + 4·23-s + 8·24-s − 4·25-s + 12·26-s − 14·27-s − 4·29-s + 4·30-s − 4·31-s + 12·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s − 0.447·5-s − 1.63·6-s − 2.26·7-s + 1.41·8-s − 9-s + 0.632·10-s + 1.80·11-s − 1.66·13-s + 3.20·14-s − 0.516·15-s − 16-s + 0.485·17-s + 1.41·18-s − 0.458·19-s − 2.61·21-s − 2.55·22-s + 0.834·23-s + 1.63·24-s − 4/5·25-s + 2.35·26-s − 2.69·27-s − 0.742·29-s + 0.730·30-s − 0.718·31-s + 2.08·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6845 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6845 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−17.4215621885, −16.8637262806, −16.7221615067, −16.0356597492, −15.2304145852, −14.8059546844, −14.2687617053, −14.0270199541, −13.0450433708, −13.0324634878, −11.9611895109, −11.7961223953, −10.8017043306, −9.81996681750, −9.75833160394, −9.10804004443, −9.03234585169, −8.40368598616, −7.59911177067, −7.11635102648, −6.31090460672, −5.44973416215, −3.99459552465, −3.50910294340, −2.56821315774, 0,
2.56821315774, 3.50910294340, 3.99459552465, 5.44973416215, 6.31090460672, 7.11635102648, 7.59911177067, 8.40368598616, 9.03234585169, 9.10804004443, 9.75833160394, 9.81996681750, 10.8017043306, 11.7961223953, 11.9611895109, 13.0324634878, 13.0450433708, 14.0270199541, 14.2687617053, 14.8059546844, 15.2304145852, 16.0356597492, 16.7221615067, 16.8637262806, 17.4215621885
