| L(s) = 1 | − 4.89·2-s − 8.92·3-s + 15.9·4-s + 43.6·6-s + 17.6·7-s − 39.0·8-s + 52.6·9-s + 11·11-s − 142.·12-s + 43.3·13-s − 86.4·14-s + 63.2·16-s + 53.1·17-s − 257.·18-s − 75.6·19-s − 157.·21-s − 53.8·22-s − 181.·23-s + 348.·24-s − 212.·26-s − 228.·27-s + 282.·28-s − 23.8·29-s + 173.·31-s + 2.59·32-s − 98.1·33-s − 260.·34-s + ⋯ |
| L(s) = 1 | − 1.73·2-s − 1.71·3-s + 1.99·4-s + 2.97·6-s + 0.953·7-s − 1.72·8-s + 1.94·9-s + 0.301·11-s − 3.42·12-s + 0.924·13-s − 1.65·14-s + 0.987·16-s + 0.758·17-s − 3.37·18-s − 0.913·19-s − 1.63·21-s − 0.521·22-s − 1.64·23-s + 2.96·24-s − 1.59·26-s − 1.63·27-s + 1.90·28-s − 0.153·29-s + 1.00·31-s + 0.0143·32-s − 0.517·33-s − 1.31·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4566374519\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4566374519\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 4.89T + 8T^{2} \) |
| 3 | \( 1 + 8.92T + 27T^{2} \) |
| 7 | \( 1 - 17.6T + 343T^{2} \) |
| 13 | \( 1 - 43.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 53.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 75.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 181.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 23.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 173.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 239.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 56.9T + 6.89e4T^{2} \) |
| 43 | \( 1 - 334.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 381.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 187.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 369.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 361.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 267.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 94.9T + 3.57e5T^{2} \) |
| 73 | \( 1 + 429.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 230.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.31e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 272.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 100.T + 9.12e5T^{2} \) |
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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
−11.41175190214076318652996487190, −10.45836313072359570918933452185, −9.854981104525944265769995063058, −8.487858840218890432582919125641, −7.73302278076985291278529148830, −6.52864226420140714098746122902, −5.84199059987235508841036261076, −4.38166770879649062919849233139, −1.74067982748143129091184094433, −0.70842764654648443474504148642,
0.70842764654648443474504148642, 1.74067982748143129091184094433, 4.38166770879649062919849233139, 5.84199059987235508841036261076, 6.52864226420140714098746122902, 7.73302278076985291278529148830, 8.487858840218890432582919125641, 9.854981104525944265769995063058, 10.45836313072359570918933452185, 11.41175190214076318652996487190
