| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 4·5-s + 4·6-s − 3·7-s − 4·8-s − 2·9-s − 8·10-s + 2·11-s − 6·12-s − 6·13-s + 6·14-s − 8·15-s + 5·16-s + 4·17-s + 4·18-s + 12·20-s + 6·21-s − 4·22-s + 23-s + 8·24-s + 6·25-s + 12·26-s + 10·27-s − 9·28-s − 4·29-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.78·5-s + 1.63·6-s − 1.13·7-s − 1.41·8-s − 2/3·9-s − 2.52·10-s + 0.603·11-s − 1.73·12-s − 1.66·13-s + 1.60·14-s − 2.06·15-s + 5/4·16-s + 0.970·17-s + 0.942·18-s + 2.68·20-s + 1.30·21-s − 0.852·22-s + 0.208·23-s + 1.63·24-s + 6/5·25-s + 2.35·26-s + 1.92·27-s − 1.70·28-s − 0.742·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2182461741\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2182461741\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−19.8583928497, −19.5800270198, −19.0835644465, −18.2120541317, −17.6722430914, −17.2128532162, −17.1574548200, −16.3370810014, −16.2140576802, −14.6077730657, −14.5961589016, −13.5383274568, −12.4227589942, −12.3052609901, −11.2313614144, −10.6815349670, −9.76554711946, −9.59481142744, −8.97193545260, −7.57571100089, −6.69441913642, −5.84432417960, −5.57928681743, −2.63232737475,
2.63232737475, 5.57928681743, 5.84432417960, 6.69441913642, 7.57571100089, 8.97193545260, 9.59481142744, 9.76554711946, 10.6815349670, 11.2313614144, 12.3052609901, 12.4227589942, 13.5383274568, 14.5961589016, 14.6077730657, 16.2140576802, 16.3370810014, 17.1574548200, 17.2128532162, 17.6722430914, 18.2120541317, 19.0835644465, 19.5800270198, 19.8583928497
