| L(s) = 1 | − 3.80·2-s + 7.34·3-s + 6.51·4-s + 5.81·5-s − 27.9·6-s − 15.8·7-s + 5.67·8-s + 26.9·9-s − 22.1·10-s − 65.8·11-s + 47.8·12-s + 72.8·13-s + 60.2·14-s + 42.6·15-s − 73.6·16-s − 39.1·17-s − 102.·18-s + 39.3·19-s + 37.8·20-s − 116.·21-s + 250.·22-s + 41.6·24-s − 91.2·25-s − 277.·26-s − 0.602·27-s − 103.·28-s − 116.·29-s + ⋯ |
| L(s) = 1 | − 1.34·2-s + 1.41·3-s + 0.813·4-s + 0.519·5-s − 1.90·6-s − 0.854·7-s + 0.250·8-s + 0.996·9-s − 0.699·10-s − 1.80·11-s + 1.15·12-s + 1.55·13-s + 1.15·14-s + 0.734·15-s − 1.15·16-s − 0.559·17-s − 1.34·18-s + 0.475·19-s + 0.422·20-s − 1.20·21-s + 2.42·22-s + 0.354·24-s − 0.729·25-s − 2.09·26-s − 0.00429·27-s − 0.695·28-s − 0.747·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 \) |
| good | 2 | \( 1 + 3.80T + 8T^{2} \) |
| 3 | \( 1 - 7.34T + 27T^{2} \) |
| 5 | \( 1 - 5.81T + 125T^{2} \) |
| 7 | \( 1 + 15.8T + 343T^{2} \) |
| 11 | \( 1 + 65.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 72.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 39.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 39.3T + 6.85e3T^{2} \) |
| 29 | \( 1 + 116.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 46.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 276.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 262.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 165.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 168.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 36.3T + 1.48e5T^{2} \) |
| 59 | \( 1 - 526.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 185.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 13.1T + 3.00e5T^{2} \) |
| 71 | \( 1 + 97.5T + 3.57e5T^{2} \) |
| 73 | \( 1 + 432.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.04e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.02e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 933.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 521.T + 9.12e5T^{2} \) |
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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
−9.859035119746643987545020141785, −8.989307242633337018414262018288, −8.419057397815698777812218068123, −7.74633060208651241329856710625, −6.75361999405841634150303298053, −5.45930288981228484619189761382, −3.74430472528369433168598958552, −2.69627495188829713304915644031, −1.69831520471135075823809365969, 0,
1.69831520471135075823809365969, 2.69627495188829713304915644031, 3.74430472528369433168598958552, 5.45930288981228484619189761382, 6.75361999405841634150303298053, 7.74633060208651241329856710625, 8.419057397815698777812218068123, 8.989307242633337018414262018288, 9.859035119746643987545020141785
