| L(s) = 1 | − 0.0323·2-s + 6.42·3-s − 7.99·4-s − 14.1·5-s − 0.207·6-s + 14.0·7-s + 0.517·8-s + 14.2·9-s + 0.456·10-s + 55.5·11-s − 51.3·12-s − 18.3·13-s − 0.453·14-s − 90.5·15-s + 63.9·16-s − 10.0·17-s − 0.460·18-s − 161.·19-s + 112.·20-s + 89.9·21-s − 1.79·22-s + 3.32·24-s + 73.8·25-s + 0.592·26-s − 81.9·27-s − 112.·28-s + 183.·29-s + ⋯ |
| L(s) = 1 | − 0.0114·2-s + 1.23·3-s − 0.999·4-s − 1.26·5-s − 0.0141·6-s + 0.756·7-s + 0.0228·8-s + 0.527·9-s + 0.0144·10-s + 1.52·11-s − 1.23·12-s − 0.390·13-s − 0.00865·14-s − 1.55·15-s + 0.999·16-s − 0.143·17-s − 0.00603·18-s − 1.94·19-s + 1.26·20-s + 0.934·21-s − 0.0174·22-s + 0.0282·24-s + 0.591·25-s + 0.00447·26-s − 0.584·27-s − 0.756·28-s + 1.17·29-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 | \( 1 + 0.0323T + 8T^{2} \) |
| 3 | \( 1 - 6.42T + 27T^{2} \) |
| 5 | \( 1 + 14.1T + 125T^{2} \) |
| 7 | \( 1 - 14.0T + 343T^{2} \) |
| 11 | \( 1 - 55.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 18.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 10.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 161.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 183.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 144.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 181.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 77.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 315.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 524.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 73.7T + 1.48e5T^{2} \) |
| 59 | \( 1 + 132.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 236.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 493.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 806.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 599.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 642.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 883.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 71.2T + 9.12e5T^{2} \) |
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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
−9.688197334496195309532537992994, −8.746627587157384368322891793473, −8.442171511007238423400662118202, −7.70173878075887255001952283561, −6.52036635073941858795456443576, −4.76045958973105609203139305775, −4.10463795098505723549789656333, −3.35459944258663402769512684220, −1.70640780935950325610822048526, 0,
1.70640780935950325610822048526, 3.35459944258663402769512684220, 4.10463795098505723549789656333, 4.76045958973105609203139305775, 6.52036635073941858795456443576, 7.70173878075887255001952283561, 8.442171511007238423400662118202, 8.746627587157384368322891793473, 9.688197334496195309532537992994
