| L(s) = 1 | − 2-s + 4-s − 4·5-s + 2·7-s − 8-s − 2·9-s + 4·10-s − 4·13-s − 2·14-s + 16-s + 4·17-s + 2·18-s − 4·20-s + 8·23-s + 6·25-s + 4·26-s + 2·28-s − 4·29-s − 32-s − 4·34-s − 8·35-s − 2·36-s − 4·37-s + 4·40-s + 4·41-s + 16·43-s + 8·45-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.78·5-s + 0.755·7-s − 0.353·8-s − 2/3·9-s + 1.26·10-s − 1.10·13-s − 0.534·14-s + 1/4·16-s + 0.970·17-s + 0.471·18-s − 0.894·20-s + 1.66·23-s + 6/5·25-s + 0.784·26-s + 0.377·28-s − 0.742·29-s − 0.176·32-s − 0.685·34-s − 1.35·35-s − 1/3·36-s − 0.657·37-s + 0.632·40-s + 0.624·41-s + 2.43·43-s + 1.19·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3130583414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3130583414\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−19.5800270198, −19.2726812580, −19.0818322220, −18.2120541317, −17.3843249773, −17.2128532162, −16.3370810014, −15.9707837644, −15.1678011465, −14.6077730657, −14.4714164826, −13.1074731398, −12.3052609901, −11.8508336106, −11.2313614144, −10.8509389842, −9.76554711946, −8.92832969723, −8.18805721451, −7.57571100089, −7.14438434722, −5.57928681743, −4.50185552371, −3.14137792992,
3.14137792992, 4.50185552371, 5.57928681743, 7.14438434722, 7.57571100089, 8.18805721451, 8.92832969723, 9.76554711946, 10.8509389842, 11.2313614144, 11.8508336106, 12.3052609901, 13.1074731398, 14.4714164826, 14.6077730657, 15.1678011465, 15.9707837644, 16.3370810014, 17.2128532162, 17.3843249773, 18.2120541317, 19.0818322220, 19.2726812580, 19.5800270198
