| L(s) = 1 | − 2.77·2-s + 2.60·3-s + 5.71·4-s − 7.24·6-s − 3.19·7-s − 10.3·8-s + 3.80·9-s − 3.35·11-s + 14.9·12-s + 1.19·13-s + 8.86·14-s + 17.2·16-s − 5.02·17-s − 10.5·18-s − 5.30·19-s − 8.32·21-s + 9.32·22-s − 5.19·23-s − 26.9·24-s − 3.30·26-s + 2.10·27-s − 18.2·28-s − 1.49·29-s + 31-s − 27.2·32-s − 8.75·33-s + 13.9·34-s + ⋯ |
| L(s) = 1 | − 1.96·2-s + 1.50·3-s + 2.85·4-s − 2.95·6-s − 1.20·7-s − 3.65·8-s + 1.26·9-s − 1.01·11-s + 4.30·12-s + 0.330·13-s + 2.36·14-s + 4.31·16-s − 1.21·17-s − 2.49·18-s − 1.21·19-s − 1.81·21-s + 1.98·22-s − 1.08·23-s − 5.49·24-s − 0.648·26-s + 0.405·27-s − 3.44·28-s − 0.276·29-s + 0.179·31-s − 4.82·32-s − 1.52·33-s + 2.39·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 2.77T + 2T^{2} \) |
| 3 | \( 1 - 2.60T + 3T^{2} \) |
| 7 | \( 1 + 3.19T + 7T^{2} \) |
| 11 | \( 1 + 3.35T + 11T^{2} \) |
| 13 | \( 1 - 1.19T + 13T^{2} \) |
| 17 | \( 1 + 5.02T + 17T^{2} \) |
| 19 | \( 1 + 5.30T + 19T^{2} \) |
| 23 | \( 1 + 5.19T + 23T^{2} \) |
| 29 | \( 1 + 1.49T + 29T^{2} \) |
| 37 | \( 1 + 5.51T + 37T^{2} \) |
| 41 | \( 1 + 0.707T + 41T^{2} \) |
| 43 | \( 1 - 4.47T + 43T^{2} \) |
| 47 | \( 1 - 8.34T + 47T^{2} \) |
| 53 | \( 1 - 1.50T + 53T^{2} \) |
| 59 | \( 1 + 0.830T + 59T^{2} \) |
| 61 | \( 1 + 9.68T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 + 4.53T + 71T^{2} \) |
| 73 | \( 1 - 5.69T + 73T^{2} \) |
| 79 | \( 1 - 8.77T + 79T^{2} \) |
| 83 | \( 1 - 7.26T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−9.623580025411262139716013895892, −8.963184127018213212081118491374, −8.418694092869295012231142605275, −7.70303164973673371988983972170, −6.84217172833091294140071711179, −6.05367167695803447657544143764, −3.74626106451345101645177675842, −2.66708122768358550046141201273, −2.07238254625535039515444698720, 0,
2.07238254625535039515444698720, 2.66708122768358550046141201273, 3.74626106451345101645177675842, 6.05367167695803447657544143764, 6.84217172833091294140071711179, 7.70303164973673371988983972170, 8.418694092869295012231142605275, 8.963184127018213212081118491374, 9.623580025411262139716013895892
