L(s) = 1 | + 1.09·2-s + 3-s − 0.809·4-s − 1.08·5-s + 1.09·6-s + 3.23·7-s − 3.06·8-s + 9-s − 1.18·10-s + 11-s − 0.809·12-s + 3.52·14-s − 1.08·15-s − 1.72·16-s + 4.11·17-s + 1.09·18-s + 3.98·19-s + 0.876·20-s + 3.23·21-s + 1.09·22-s − 2.86·23-s − 3.06·24-s − 3.82·25-s + 27-s − 2.61·28-s + 5.17·29-s − 1.18·30-s + ⋯ |
L(s) = 1 | + 0.771·2-s + 0.577·3-s − 0.404·4-s − 0.484·5-s + 0.445·6-s + 1.22·7-s − 1.08·8-s + 0.333·9-s − 0.373·10-s + 0.301·11-s − 0.233·12-s + 0.942·14-s − 0.279·15-s − 0.431·16-s + 0.997·17-s + 0.257·18-s + 0.914·19-s + 0.196·20-s + 0.705·21-s + 0.232·22-s − 0.596·23-s − 0.625·24-s − 0.765·25-s + 0.192·27-s − 0.494·28-s + 0.960·29-s − 0.215·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.366556192\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.366556192\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 1.09T + 2T^{2} \) |
| 5 | \( 1 + 1.08T + 5T^{2} \) |
| 7 | \( 1 - 3.23T + 7T^{2} \) |
| 17 | \( 1 - 4.11T + 17T^{2} \) |
| 19 | \( 1 - 3.98T + 19T^{2} \) |
| 23 | \( 1 + 2.86T + 23T^{2} \) |
| 29 | \( 1 - 5.17T + 29T^{2} \) |
| 31 | \( 1 + 8.59T + 31T^{2} \) |
| 37 | \( 1 - 9.54T + 37T^{2} \) |
| 41 | \( 1 + 6.65T + 41T^{2} \) |
| 43 | \( 1 - 2.42T + 43T^{2} \) |
| 47 | \( 1 + 5.83T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 - 0.565T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 1.87T + 79T^{2} \) |
| 83 | \( 1 - 2.40T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−8.116937145382469193125200737692, −7.63833781308686224995995007614, −6.73962742397131752692307627248, −5.62508397367515071854637454718, −5.23994380194397900849691606134, −4.32478029438611561265551980390, −3.83339533061073695988555271179, −3.09726297359940938767779300502, −2.01245136062505317762436284488, −0.884746829601307619181461317678,
0.884746829601307619181461317678, 2.01245136062505317762436284488, 3.09726297359940938767779300502, 3.83339533061073695988555271179, 4.32478029438611561265551980390, 5.23994380194397900849691606134, 5.62508397367515071854637454718, 6.73962742397131752692307627248, 7.63833781308686224995995007614, 8.116937145382469193125200737692