| L(s) = 1 | − 4-s − 9-s − 8·11-s + 16-s − 8·19-s + 4·29-s + 36-s − 12·41-s + 8·44-s − 49-s − 24·59-s + 28·61-s − 64-s − 16·71-s + 8·76-s − 32·79-s + 81-s − 20·89-s + 8·99-s + 12·101-s − 28·109-s − 4·116-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s − 2.41·11-s + 1/4·16-s − 1.83·19-s + 0.742·29-s + 1/6·36-s − 1.87·41-s + 1.20·44-s − 1/7·49-s − 3.12·59-s + 3.58·61-s − 1/8·64-s − 1.89·71-s + 0.917·76-s − 3.60·79-s + 1/9·81-s − 2.11·89-s + 0.804·99-s + 1.19·101-s − 2.68·109-s − 0.371·116-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2970836594\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2970836594\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−10.20977089497244142164150412540, −9.871803531896670895488597185168, −9.298537464081603216212391497777, −8.639645902865867538134333736917, −8.370954911433617554404760448593, −8.311706994511949254366048060545, −7.76958854243572698548234169468, −7.09861969659027575895413849968, −6.95129206243967525854584535116, −6.15420714896632392776529592794, −5.85525530889552964967022684095, −5.26263479452017392656278536065, −5.06175895540358495488219928105, −4.28934501251681630997564783480, −4.24045980674624965924177980555, −3.04483606086810999659184355766, −3.01078274682895516553064085863, −2.27374402152380305517877193771, −1.60551894362912374085680735087, −0.23642560944077347037438849255,
0.23642560944077347037438849255, 1.60551894362912374085680735087, 2.27374402152380305517877193771, 3.01078274682895516553064085863, 3.04483606086810999659184355766, 4.24045980674624965924177980555, 4.28934501251681630997564783480, 5.06175895540358495488219928105, 5.26263479452017392656278536065, 5.85525530889552964967022684095, 6.15420714896632392776529592794, 6.95129206243967525854584535116, 7.09861969659027575895413849968, 7.76958854243572698548234169468, 8.311706994511949254366048060545, 8.370954911433617554404760448593, 8.639645902865867538134333736917, 9.298537464081603216212391497777, 9.871803531896670895488597185168, 10.20977089497244142164150412540
