| L(s) = 1 | − 2·3-s − 3·5-s + 2·7-s − 2·9-s + 8·13-s + 6·15-s − 4·17-s − 4·19-s − 4·21-s + 6·23-s + 2·25-s + 10·27-s − 4·29-s − 4·31-s − 6·35-s − 16·39-s + 16·41-s − 10·43-s + 6·45-s − 6·47-s − 10·49-s + 8·51-s + 8·53-s + 8·57-s + 12·59-s − 8·61-s − 4·63-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.34·5-s + 0.755·7-s − 2/3·9-s + 2.21·13-s + 1.54·15-s − 0.970·17-s − 0.917·19-s − 0.872·21-s + 1.25·23-s + 2/5·25-s + 1.92·27-s − 0.742·29-s − 0.718·31-s − 1.01·35-s − 2.56·39-s + 2.49·41-s − 1.52·43-s + 0.894·45-s − 0.875·47-s − 1.42·49-s + 1.12·51-s + 1.09·53-s + 1.05·57-s + 1.56·59-s − 1.02·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3085697573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3085697573\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−19.6035760866, −19.1887245783, −18.1271877078, −18.1183629461, −17.3081878781, −16.7385636699, −16.0995603957, −15.7488207479, −14.7919331095, −14.5765256398, −13.2768755254, −13.0023664158, −11.7666126827, −11.5085445320, −10.9076921437, −10.8347395030, −8.95538623117, −8.55221755078, −7.77199473906, −6.57891116466, −5.87146418849, −4.78130792718, −3.67478222653,
3.67478222653, 4.78130792718, 5.87146418849, 6.57891116466, 7.77199473906, 8.55221755078, 8.95538623117, 10.8347395030, 10.9076921437, 11.5085445320, 11.7666126827, 13.0023664158, 13.2768755254, 14.5765256398, 14.7919331095, 15.7488207479, 16.0995603957, 16.7385636699, 17.3081878781, 18.1183629461, 18.1271877078, 19.1887245783, 19.6035760866
