L(s) = 1 | − 2-s + 3-s − 4-s − 4·5-s − 6-s − 7-s + 3·8-s + 9-s + 4·10-s + 4·11-s − 12-s − 8·13-s + 14-s − 4·15-s − 16-s − 8·17-s − 18-s + 4·19-s + 4·20-s − 21-s − 4·22-s + 3·24-s + 2·25-s + 8·26-s + 27-s + 28-s − 12·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 1.78·5-s − 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s + 1.26·10-s + 1.20·11-s − 0.288·12-s − 2.21·13-s + 0.267·14-s − 1.03·15-s − 1/4·16-s − 1.94·17-s − 0.235·18-s + 0.917·19-s + 0.894·20-s − 0.218·21-s − 0.852·22-s + 0.612·24-s + 2/5·25-s + 1.56·26-s + 0.192·27-s + 0.188·28-s − 2.22·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12096 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−16.7387181439, −15.8734598001, −15.8703035732, −15.0146850926, −14.6798782297, −14.5135323404, −13.4880567789, −13.1937925053, −12.7188757920, −11.8432236791, −11.5559508175, −11.3628128906, −10.2338342589, −9.72466121073, −9.46876170590, −8.67236519668, −8.43945012578, −7.52766744710, −7.25047783803, −6.93944627141, −5.54216083420, −4.56897038264, −4.13559084051, −3.55808048735, −2.19360926670, 0,
2.19360926670, 3.55808048735, 4.13559084051, 4.56897038264, 5.54216083420, 6.93944627141, 7.25047783803, 7.52766744710, 8.43945012578, 8.67236519668, 9.46876170590, 9.72466121073, 10.2338342589, 11.3628128906, 11.5559508175, 11.8432236791, 12.7188757920, 13.1937925053, 13.4880567789, 14.5135323404, 14.6798782297, 15.0146850926, 15.8703035732, 15.8734598001, 16.7387181439