| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s + 2·7-s − 8-s + 3·9-s − 10·11-s − 2·12-s − 4·13-s − 2·14-s + 16-s − 6·17-s − 3·18-s + 4·19-s − 4·21-s + 10·22-s − 23-s + 2·24-s − 6·25-s + 4·26-s − 4·27-s + 2·28-s + 8·29-s + 4·31-s − 32-s + 20·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.755·7-s − 0.353·8-s + 9-s − 3.01·11-s − 0.577·12-s − 1.10·13-s − 0.534·14-s + 1/4·16-s − 1.45·17-s − 0.707·18-s + 0.917·19-s − 0.872·21-s + 2.13·22-s − 0.208·23-s + 0.408·24-s − 6/5·25-s + 0.784·26-s − 0.769·27-s + 0.377·28-s + 1.48·29-s + 0.718·31-s − 0.176·32-s + 3.48·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13248 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13248 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−16.2846034524, −15.9834358984, −15.8931893486, −15.0896927147, −14.9623475756, −13.8794868848, −13.4201720335, −13.0286277271, −12.4460165753, −11.6752538533, −11.6154866441, −10.8484944274, −10.4895217551, −9.83286035817, −9.82663766019, −8.39104653369, −8.25912428821, −7.48954786032, −7.17212008873, −6.25250569927, −5.56983356276, −4.78570582428, −4.77602747707, −2.91631229938, −2.09803110754, 0,
2.09803110754, 2.91631229938, 4.77602747707, 4.78570582428, 5.56983356276, 6.25250569927, 7.17212008873, 7.48954786032, 8.25912428821, 8.39104653369, 9.82663766019, 9.83286035817, 10.4895217551, 10.8484944274, 11.6154866441, 11.6752538533, 12.4460165753, 13.0286277271, 13.4201720335, 13.8794868848, 14.9623475756, 15.0896927147, 15.8931893486, 15.9834358984, 16.2846034524
