| L(s) = 1 | + 2·3-s + 3·9-s − 4·11-s + 2·13-s + 4·17-s + 8·23-s − 2·25-s + 4·27-s − 4·29-s + 8·31-s − 8·33-s + 4·37-s + 4·39-s + 16·41-s + 8·43-s + 12·47-s − 6·49-s + 8·51-s + 4·53-s + 4·59-s − 4·61-s + 8·67-s + 16·69-s − 4·71-s + 12·73-s − 4·75-s + 5·81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s − 1.20·11-s + 0.554·13-s + 0.970·17-s + 1.66·23-s − 2/5·25-s + 0.769·27-s − 0.742·29-s + 1.43·31-s − 1.39·33-s + 0.657·37-s + 0.640·39-s + 2.49·41-s + 1.21·43-s + 1.75·47-s − 6/7·49-s + 1.12·51-s + 0.549·53-s + 0.520·59-s − 0.512·61-s + 0.977·67-s + 1.92·69-s − 0.474·71-s + 1.40·73-s − 0.461·75-s + 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.121278315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.121278315\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 16 T + 138 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 24 T + 314 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.330148198629625594760511962355, −8.692630339027025144046279340810, −8.145080102630749929089990402603, −8.128273444280305597521551857007, −7.57174069222192858537738023107, −7.46041064689463067461640259431, −6.95783076853158416913785270033, −6.48287311677572367411051001991, −5.87407383893272497913253718580, −5.69739223489035712162797776841, −5.18307125862536508612877123679, −4.69587042821263142397230170436, −4.20017612345135578446174577754, −3.90658987269524516933528121477, −3.20065359825704428798410372397, −3.00475806564006354845362230968, −2.33284347810077855114796179112, −2.25024397012376614252460378200, −1.05079080296021778812399079345, −0.877497724290521036202874339273,
0.877497724290521036202874339273, 1.05079080296021778812399079345, 2.25024397012376614252460378200, 2.33284347810077855114796179112, 3.00475806564006354845362230968, 3.20065359825704428798410372397, 3.90658987269524516933528121477, 4.20017612345135578446174577754, 4.69587042821263142397230170436, 5.18307125862536508612877123679, 5.69739223489035712162797776841, 5.87407383893272497913253718580, 6.48287311677572367411051001991, 6.95783076853158416913785270033, 7.46041064689463067461640259431, 7.57174069222192858537738023107, 8.128273444280305597521551857007, 8.145080102630749929089990402603, 8.692630339027025144046279340810, 9.330148198629625594760511962355
