| L(s) = 1 | − 3·3-s − 2·5-s − 7-s + 6·9-s + 11-s + 6·13-s + 6·15-s − 10·17-s + 14·19-s + 3·21-s + 4·23-s + 5·25-s − 9·27-s + 4·29-s − 6·31-s − 3·33-s + 2·35-s + 4·37-s − 18·39-s − 3·41-s − 43-s − 12·45-s + 30·51-s + 24·53-s − 2·55-s − 42·57-s − 7·59-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.894·5-s − 0.377·7-s + 2·9-s + 0.301·11-s + 1.66·13-s + 1.54·15-s − 2.42·17-s + 3.21·19-s + 0.654·21-s + 0.834·23-s + 25-s − 1.73·27-s + 0.742·29-s − 1.07·31-s − 0.522·33-s + 0.338·35-s + 0.657·37-s − 2.88·39-s − 0.468·41-s − 0.152·43-s − 1.78·45-s + 4.20·51-s + 3.29·53-s − 0.269·55-s − 5.56·57-s − 0.911·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.138363842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.138363842\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 7 T - 10 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 6 T - 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 16 T + 173 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.36888140826630746176488953348, −9.750602134832573164493971686400, −9.390380442065762142082684866181, −9.024756521375990362504658290706, −8.437375492676719228509422488476, −8.232515500873521451287137570030, −7.23215565870312661102432643279, −7.21114181375660063017265460879, −6.77451638415183494774830382539, −6.47773240772197719564003062171, −5.72133422824476750017118463218, −5.59414197443045563636132795290, −4.88781362742784228309991891096, −4.69645324057902840504579966857, −3.80716737012656742753774995804, −3.74294154192593674966189724795, −3.02128274132811354066849985770, −2.07452330142743886164965910853, −0.906756118738555286380305438060, −0.812235298555252510886035275990,
0.812235298555252510886035275990, 0.906756118738555286380305438060, 2.07452330142743886164965910853, 3.02128274132811354066849985770, 3.74294154192593674966189724795, 3.80716737012656742753774995804, 4.69645324057902840504579966857, 4.88781362742784228309991891096, 5.59414197443045563636132795290, 5.72133422824476750017118463218, 6.47773240772197719564003062171, 6.77451638415183494774830382539, 7.21114181375660063017265460879, 7.23215565870312661102432643279, 8.232515500873521451287137570030, 8.437375492676719228509422488476, 9.024756521375990362504658290706, 9.390380442065762142082684866181, 9.750602134832573164493971686400, 10.36888140826630746176488953348
