| L(s) = 1 | − 2·3-s − 2·7-s − 2·9-s + 4·11-s − 4·17-s + 4·21-s + 10·23-s + 25-s + 10·27-s + 4·29-s − 12·31-s − 8·33-s + 8·37-s − 18·43-s − 2·47-s − 2·49-s + 8·51-s + 8·59-s + 4·63-s + 10·67-s − 20·69-s − 12·71-s − 4·73-s − 2·75-s − 8·77-s + 8·79-s − 5·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.755·7-s − 2/3·9-s + 1.20·11-s − 0.970·17-s + 0.872·21-s + 2.08·23-s + 1/5·25-s + 1.92·27-s + 0.742·29-s − 2.15·31-s − 1.39·33-s + 1.31·37-s − 2.74·43-s − 0.291·47-s − 2/7·49-s + 1.12·51-s + 1.04·59-s + 0.503·63-s + 1.22·67-s − 2.40·69-s − 1.42·71-s − 0.468·73-s − 0.230·75-s − 0.911·77-s + 0.900·79-s − 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3493781408\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3493781408\) |
| \(L(\frac{3}{2})\) |
|
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_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 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$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−19.6420667034, −19.6035760866, −18.4963041394, −18.1271877078, −17.3081878781, −17.0080103482, −16.5436791565, −16.0995603957, −14.9509208965, −14.7919331095, −13.9008699252, −13.0023664158, −12.7095155923, −11.5242621401, −11.5085445320, −10.8347395030, −9.85184018389, −9.02203563842, −8.55221755078, −6.97238544391, −6.57891116466, −5.70093888867, −4.78130792718, −3.26180587408,
3.26180587408, 4.78130792718, 5.70093888867, 6.57891116466, 6.97238544391, 8.55221755078, 9.02203563842, 9.85184018389, 10.8347395030, 11.5085445320, 11.5242621401, 12.7095155923, 13.0023664158, 13.9008699252, 14.7919331095, 14.9509208965, 16.0995603957, 16.5436791565, 17.0080103482, 17.3081878781, 18.1271877078, 18.4963041394, 19.6035760866, 19.6420667034
