| L(s) = 1 | − 5.60·2-s + 3·3-s + 23.4·4-s + 12.9·5-s − 16.8·6-s − 18.2·7-s − 86.4·8-s + 9·9-s − 72.7·10-s − 71.8·11-s + 70.2·12-s + 31.4·13-s + 102.·14-s + 38.9·15-s + 297.·16-s − 94.8·17-s − 50.4·18-s − 53.7·19-s + 303.·20-s − 54.8·21-s + 402.·22-s − 70.2·23-s − 259.·24-s + 43.3·25-s − 176.·26-s + 27·27-s − 428.·28-s + ⋯ |
| L(s) = 1 | − 1.98·2-s + 0.577·3-s + 2.92·4-s + 1.16·5-s − 1.14·6-s − 0.988·7-s − 3.81·8-s + 0.333·9-s − 2.29·10-s − 1.96·11-s + 1.69·12-s + 0.671·13-s + 1.95·14-s + 0.670·15-s + 4.64·16-s − 1.35·17-s − 0.660·18-s − 0.648·19-s + 3.39·20-s − 0.570·21-s + 3.90·22-s − 0.636·23-s − 2.20·24-s + 0.346·25-s − 1.33·26-s + 0.192·27-s − 2.89·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 3T \) |
| 59 | \( 1 + 59T \) |
| good | 2 | \( 1 + 5.60T + 8T^{2} \) |
| 5 | \( 1 - 12.9T + 125T^{2} \) |
| 7 | \( 1 + 18.2T + 343T^{2} \) |
| 11 | \( 1 + 71.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 31.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 94.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 53.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 70.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 20.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 5.95T + 2.97e4T^{2} \) |
| 37 | \( 1 - 167.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 248.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 472.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 140.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 21.3T + 1.48e5T^{2} \) |
| 61 | \( 1 - 375.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 410.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 420.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 865.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 53.7T + 4.93e5T^{2} \) |
| 83 | \( 1 - 39.4T + 5.71e5T^{2} \) |
| 89 | \( 1 + 354.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.41e3T + 9.12e5T^{2} \) |
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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
−11.10677280328102982850931961541, −10.25899702626271319053210031752, −9.743886129165784979380739931287, −8.784126385570542667758491947412, −7.981171944413546645303115251282, −6.73248434856157670572974565095, −5.90033753096494836344236706515, −2.84573739504704145327438329099, −2.01057895566647514118964697868, 0,
2.01057895566647514118964697868, 2.84573739504704145327438329099, 5.90033753096494836344236706515, 6.73248434856157670572974565095, 7.981171944413546645303115251282, 8.784126385570542667758491947412, 9.743886129165784979380739931287, 10.25899702626271319053210031752, 11.10677280328102982850931961541
