| L(s) = 1 | − 1.48·2-s − 6.79·3-s − 5.80·4-s − 11.4·5-s + 10.0·6-s − 21.1·7-s + 20.4·8-s + 19.1·9-s + 17.0·10-s − 42.5·11-s + 39.4·12-s + 64.5·13-s + 31.3·14-s + 78.0·15-s + 16.0·16-s − 40.8·17-s − 28.3·18-s + 102.·19-s + 66.6·20-s + 143.·21-s + 63.0·22-s − 138.·24-s + 6.99·25-s − 95.7·26-s + 53.4·27-s + 122.·28-s − 89.8·29-s + ⋯ |
| L(s) = 1 | − 0.524·2-s − 1.30·3-s − 0.725·4-s − 1.02·5-s + 0.685·6-s − 1.14·7-s + 0.904·8-s + 0.708·9-s + 0.538·10-s − 1.16·11-s + 0.947·12-s + 1.37·13-s + 0.598·14-s + 1.34·15-s + 0.251·16-s − 0.583·17-s − 0.371·18-s + 1.23·19-s + 0.745·20-s + 1.49·21-s + 0.611·22-s − 1.18·24-s + 0.0559·25-s − 0.721·26-s + 0.380·27-s + 0.827·28-s − 0.575·29-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 | \( 1 + 1.48T + 8T^{2} \) |
| 3 | \( 1 + 6.79T + 27T^{2} \) |
| 5 | \( 1 + 11.4T + 125T^{2} \) |
| 7 | \( 1 + 21.1T + 343T^{2} \) |
| 11 | \( 1 + 42.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 64.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 40.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 102.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 89.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 232.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 91.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 84.5T + 6.89e4T^{2} \) |
| 43 | \( 1 - 44.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 326.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 11.1T + 1.48e5T^{2} \) |
| 59 | \( 1 + 768.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 772.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.07e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 226.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 698.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 355.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 281.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 256.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.05e3T + 9.12e5T^{2} \) |
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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
−10.16615591480876942307490006331, −9.159683635025173740834684322508, −8.199185396571785238067449424456, −7.35381144314991121101675246902, −6.23300965434008873804308440869, −5.36688534656834850615249302668, −4.31609804541031134256280525583, −3.26537719855639006275675021616, −0.824610559964219805676865204165, 0,
0.824610559964219805676865204165, 3.26537719855639006275675021616, 4.31609804541031134256280525583, 5.36688534656834850615249302668, 6.23300965434008873804308440869, 7.35381144314991121101675246902, 8.199185396571785238067449424456, 9.159683635025173740834684322508, 10.16615591480876942307490006331
