| L(s) = 1 | − 3.91·2-s + 5.59·3-s + 11.3·4-s − 21.9·6-s − 7·7-s − 28.6·8-s + 22.3·9-s + 63.3·12-s + 10.6·13-s + 27.4·14-s + 66.9·16-s + 8.62·17-s − 87.3·18-s + 15.6·19-s − 39.1·21-s − 8.23·23-s − 160.·24-s + 25·25-s − 41.5·26-s + 74.4·27-s − 79.2·28-s − 147.·32-s − 33.7·34-s + 252.·36-s + 73.2·37-s − 61.2·38-s + 59.3·39-s + ⋯ |
| L(s) = 1 | − 1.95·2-s + 1.86·3-s + 2.83·4-s − 3.65·6-s − 7-s − 3.58·8-s + 2.47·9-s + 5.27·12-s + 0.815·13-s + 1.95·14-s + 4.18·16-s + 0.507·17-s − 4.85·18-s + 0.823·19-s − 1.86·21-s − 0.358·23-s − 6.68·24-s + 25-s − 1.59·26-s + 2.75·27-s − 2.83·28-s − 4.60·32-s − 0.993·34-s + 7.01·36-s + 1.97·37-s − 1.61·38-s + 1.52·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.292869531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.292869531\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 7T \) |
| 41 | \( 1 - 41T \) |
| good | 2 | \( 1 + 3.91T + 4T^{2} \) |
| 3 | \( 1 - 5.59T + 9T^{2} \) |
| 5 | \( 1 - 25T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 - 10.6T + 169T^{2} \) |
| 17 | \( 1 - 8.62T + 289T^{2} \) |
| 19 | \( 1 - 15.6T + 361T^{2} \) |
| 23 | \( 1 + 8.23T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 73.2T + 1.36e3T^{2} \) |
| 43 | \( 1 + 81.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 83.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 + 42.9T + 7.92e3T^{2} \) |
| 97 | \( 1 + 159.T + 9.40e3T^{2} \) |
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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
−11.15031704938156029830607267766, −9.892287472992093845180107027767, −9.683551748889042503636885749605, −8.741342764493943674266833724058, −8.096774639782781527990756117318, −7.26111900519273187048898808640, −6.33231681632397500224251090770, −3.44833566716796602454560871823, −2.69269007198505005667560617691, −1.26334414844324904248444218967,
1.26334414844324904248444218967, 2.69269007198505005667560617691, 3.44833566716796602454560871823, 6.33231681632397500224251090770, 7.26111900519273187048898808640, 8.096774639782781527990756117318, 8.741342764493943674266833724058, 9.683551748889042503636885749605, 9.892287472992093845180107027767, 11.15031704938156029830607267766
