L(s) = 1 | + 3.15·2-s + 1.99·3-s + 1.94·4-s − 5.00·5-s + 6.27·6-s + 2.78·7-s − 19.1·8-s − 23.0·9-s − 15.7·10-s − 27.2·11-s + 3.86·12-s − 59.7·13-s + 8.78·14-s − 9.97·15-s − 75.7·16-s − 72.6·18-s + 33.2·19-s − 9.71·20-s + 5.54·21-s − 85.9·22-s + 210.·23-s − 38.0·24-s − 99.9·25-s − 188.·26-s − 99.6·27-s + 5.40·28-s − 20.0·29-s + ⋯ |
L(s) = 1 | + 1.11·2-s + 0.383·3-s + 0.242·4-s − 0.447·5-s + 0.427·6-s + 0.150·7-s − 0.844·8-s − 0.853·9-s − 0.499·10-s − 0.746·11-s + 0.0929·12-s − 1.27·13-s + 0.167·14-s − 0.171·15-s − 1.18·16-s − 0.951·18-s + 0.401·19-s − 0.108·20-s + 0.0576·21-s − 0.832·22-s + 1.91·23-s − 0.323·24-s − 0.799·25-s − 1.42·26-s − 0.710·27-s + 0.0364·28-s − 0.128·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 \) |
good | 2 | \( 1 - 3.15T + 8T^{2} \) |
| 3 | \( 1 - 1.99T + 27T^{2} \) |
| 5 | \( 1 + 5.00T + 125T^{2} \) |
| 7 | \( 1 - 2.78T + 343T^{2} \) |
| 11 | \( 1 + 27.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 59.7T + 2.19e3T^{2} \) |
| 19 | \( 1 - 33.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 210.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 20.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 133.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 152.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 261.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 316.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 329.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 310.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 54.7T + 2.05e5T^{2} \) |
| 61 | \( 1 - 818.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 731.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 629.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 496.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 90.0T + 4.93e5T^{2} \) |
| 83 | \( 1 - 364.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 192.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.35e3T + 9.12e5T^{2} \) |
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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
−11.30851947879964036174815772891, −9.946745493332234891950591065366, −8.902689968101943823630969475237, −7.964466508431045695325469645968, −6.85302170543855950139595440634, −5.41759693288690233451686978426, −4.83525916146163783885984894187, −3.43103592106447743886612754842, −2.58662334850458093131607388487, 0,
2.58662334850458093131607388487, 3.43103592106447743886612754842, 4.83525916146163783885984894187, 5.41759693288690233451686978426, 6.85302170543855950139595440634, 7.964466508431045695325469645968, 8.902689968101943823630969475237, 9.946745493332234891950591065366, 11.30851947879964036174815772891