| L(s) = 1 | + 4.24·2-s + 9.99·4-s + 13.9·5-s + 12.9·7-s + 8.48·8-s + 59.2·10-s − 49.9·11-s + 32.9·13-s + 55.0·14-s − 44.0·16-s + 17·17-s + 54.8·19-s + 139.·20-s − 212.·22-s + 82.0·23-s + 70.1·25-s + 139.·26-s + 129.·28-s − 289.·29-s + 232.·31-s − 254.·32-s + 72.1·34-s + 181.·35-s − 227.·37-s + 232.·38-s + 118.·40-s − 437.·41-s + ⋯ |
| L(s) = 1 | + 1.49·2-s + 1.24·4-s + 1.24·5-s + 0.700·7-s + 0.374·8-s + 1.87·10-s − 1.36·11-s + 0.702·13-s + 1.05·14-s − 0.687·16-s + 0.242·17-s + 0.662·19-s + 1.56·20-s − 2.05·22-s + 0.743·23-s + 0.561·25-s + 1.05·26-s + 0.875·28-s − 1.85·29-s + 1.34·31-s − 1.40·32-s + 0.363·34-s + 0.875·35-s − 1.01·37-s + 0.994·38-s + 0.468·40-s − 1.66·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.295824191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.295824191\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 2 | \( 1 - 4.24T + 8T^{2} \) |
| 5 | \( 1 - 13.9T + 125T^{2} \) |
| 7 | \( 1 - 12.9T + 343T^{2} \) |
| 11 | \( 1 + 49.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 32.9T + 2.19e3T^{2} \) |
| 19 | \( 1 - 54.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 82.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 289.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 232.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 227.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 437.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 158.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 159.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 376.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 185.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 861.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 178.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.16e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 383.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 254T + 4.93e5T^{2} \) |
| 83 | \( 1 + 447.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.21e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 291.T + 9.12e5T^{2} \) |
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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
−12.99994509698210638005892789033, −11.70354962673215597359490137576, −10.76633119057182824225530895573, −9.630808692506768773583318036634, −8.204045277479455559080986788716, −6.71381677280728321624757065067, −5.48642687970852796293955126988, −5.05954334559111602843225664696, −3.31471418618607484743961164050, −1.95985610442426497159240132776,
1.95985610442426497159240132776, 3.31471418618607484743961164050, 5.05954334559111602843225664696, 5.48642687970852796293955126988, 6.71381677280728321624757065067, 8.204045277479455559080986788716, 9.630808692506768773583318036634, 10.76633119057182824225530895573, 11.70354962673215597359490137576, 12.99994509698210638005892789033
