| L(s) = 1 | − 2·3-s + 4·7-s + 3·9-s − 4·13-s − 4·17-s − 8·21-s − 8·23-s − 6·25-s − 4·27-s + 8·29-s + 4·31-s + 4·37-s + 8·39-s − 4·41-s − 8·47-s + 2·49-s + 8·51-s + 8·53-s + 8·59-s + 4·61-s + 12·63-s − 8·67-s + 16·69-s + 24·71-s + 4·73-s + 12·75-s − 12·79-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.51·7-s + 9-s − 1.10·13-s − 0.970·17-s − 1.74·21-s − 1.66·23-s − 6/5·25-s − 0.769·27-s + 1.48·29-s + 0.718·31-s + 0.657·37-s + 1.28·39-s − 0.624·41-s − 1.16·47-s + 2/7·49-s + 1.12·51-s + 1.09·53-s + 1.04·59-s + 0.512·61-s + 1.51·63-s − 0.977·67-s + 1.92·69-s + 2.84·71-s + 0.468·73-s + 1.38·75-s − 1.35·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5397097588\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5397097588\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−18.2733510574, −17.9221722496, −17.6707519500, −17.2258552513, −16.6272610009, −15.9834358984, −15.4732516475, −14.9623475756, −14.1819882616, −13.8794868848, −13.0286277271, −12.3172568607, −11.6752538533, −11.5820474614, −10.8484944274, −9.96467191573, −9.82663766019, −8.39104653369, −8.09899069409, −7.17212008873, −6.42897107073, −5.56983356276, −4.77602747707, −4.25303028693, −2.09803110754,
2.09803110754, 4.25303028693, 4.77602747707, 5.56983356276, 6.42897107073, 7.17212008873, 8.09899069409, 8.39104653369, 9.82663766019, 9.96467191573, 10.8484944274, 11.5820474614, 11.6752538533, 12.3172568607, 13.0286277271, 13.8794868848, 14.1819882616, 14.9623475756, 15.4732516475, 15.9834358984, 16.6272610009, 17.2258552513, 17.6707519500, 17.9221722496, 18.2733510574
