L(s) = 1 | − 1.80·2-s − 0.445·3-s + 2.24·4-s − 1.80·5-s + 0.801·6-s − 2.24·8-s − 0.801·9-s + 3.24·10-s + 2·11-s − 12-s + 0.801·15-s + 1.80·16-s + 1.44·18-s − 0.445·19-s − 4.04·20-s − 3.60·22-s + 1.24·23-s + 1.00·24-s + 2.24·25-s + 0.801·27-s + 1.24·29-s − 1.44·30-s − 1.00·32-s − 0.890·33-s − 1.80·36-s + 0.801·38-s + 4.04·40-s + ⋯ |
L(s) = 1 | − 1.80·2-s − 0.445·3-s + 2.24·4-s − 1.80·5-s + 0.801·6-s − 2.24·8-s − 0.801·9-s + 3.24·10-s + 2·11-s − 12-s + 0.801·15-s + 1.80·16-s + 1.44·18-s − 0.445·19-s − 4.04·20-s − 3.60·22-s + 1.24·23-s + 1.00·24-s + 2.24·25-s + 0.801·27-s + 1.24·29-s − 1.44·30-s − 1.00·32-s − 0.890·33-s − 1.80·36-s + 0.801·38-s + 4.04·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2486380949\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2486380949\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 431 | \( 1 - T \) |
good | 2 | \( 1 + 1.80T + T^{2} \) |
| 3 | \( 1 + 0.445T + T^{2} \) |
| 5 | \( 1 + 1.80T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - 2T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 0.445T + T^{2} \) |
| 23 | \( 1 - 1.24T + T^{2} \) |
| 29 | \( 1 - 1.24T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.24T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 0.445T + T^{2} \) |
| 59 | \( 1 + 1.80T + T^{2} \) |
| 61 | \( 1 + 0.445T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - 1.24T + T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.19228054737707333273482427874, −10.68712987451900361098159070559, −9.200620306338118211942704328064, −8.762861074425388030506493418900, −7.916432586930642498200672366598, −6.99697122923954466430808933502, −6.31790254194168164270036254696, −4.39876310194177161529601183170, −3.08646167109651913768329269533, −0.948072856839569317356302017586,
0.948072856839569317356302017586, 3.08646167109651913768329269533, 4.39876310194177161529601183170, 6.31790254194168164270036254696, 6.99697122923954466430808933502, 7.916432586930642498200672366598, 8.762861074425388030506493418900, 9.200620306338118211942704328064, 10.68712987451900361098159070559, 11.19228054737707333273482427874