| L(s) = 1 | − 0.667·2-s + 2.16·3-s − 1.55·4-s + 3.76·5-s − 1.44·6-s − 2.15·7-s + 2.37·8-s + 1.67·9-s − 2.51·10-s + 11-s − 3.36·12-s − 3.37·13-s + 1.43·14-s + 8.15·15-s + 1.52·16-s + 6.32·17-s − 1.11·18-s − 3.38·19-s − 5.85·20-s − 4.65·21-s − 0.667·22-s − 2.58·23-s + 5.13·24-s + 9.21·25-s + 2.25·26-s − 2.86·27-s + 3.34·28-s + ⋯ |
| L(s) = 1 | − 0.472·2-s + 1.24·3-s − 0.776·4-s + 1.68·5-s − 0.589·6-s − 0.814·7-s + 0.839·8-s + 0.558·9-s − 0.796·10-s + 0.301·11-s − 0.969·12-s − 0.935·13-s + 0.384·14-s + 2.10·15-s + 0.380·16-s + 1.53·17-s − 0.263·18-s − 0.776·19-s − 1.30·20-s − 1.01·21-s − 0.142·22-s − 0.539·23-s + 1.04·24-s + 1.84·25-s + 0.441·26-s − 0.551·27-s + 0.632·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 0.667T + 2T^{2} \) |
| 3 | \( 1 - 2.16T + 3T^{2} \) |
| 5 | \( 1 - 3.76T + 5T^{2} \) |
| 7 | \( 1 + 2.15T + 7T^{2} \) |
| 13 | \( 1 + 3.37T + 13T^{2} \) |
| 17 | \( 1 - 6.32T + 17T^{2} \) |
| 19 | \( 1 + 3.38T + 19T^{2} \) |
| 23 | \( 1 + 2.58T + 23T^{2} \) |
| 31 | \( 1 + 7.71T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 - 4.34T + 41T^{2} \) |
| 43 | \( 1 + 3.26T + 43T^{2} \) |
| 47 | \( 1 + 7.48T + 47T^{2} \) |
| 53 | \( 1 + 5.54T + 53T^{2} \) |
| 59 | \( 1 - 3.76T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 + 2.13T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 5.82T + 83T^{2} \) |
| 89 | \( 1 - 9.99T + 89T^{2} \) |
| 97 | \( 1 + 6.92T + 97T^{2} \) |
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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
−7.56487692956172949258530684616, −6.82102385652941119530229738124, −5.93482885258334875495046325271, −5.41060134977255617235148670281, −4.59058041122269390813023720490, −3.50973602416323743881992438884, −3.08148799173426171987502422241, −2.01308242627617501068426495161, −1.55160556835984579396574377579, 0,
1.55160556835984579396574377579, 2.01308242627617501068426495161, 3.08148799173426171987502422241, 3.50973602416323743881992438884, 4.59058041122269390813023720490, 5.41060134977255617235148670281, 5.93482885258334875495046325271, 6.82102385652941119530229738124, 7.56487692956172949258530684616
