| L(s) = 1 | + 9.20·2-s + 9·3-s + 52.8·4-s + 82.8·6-s − 195.·7-s + 191.·8-s + 81·9-s + 121·11-s + 475.·12-s − 457.·13-s − 1.79e3·14-s + 74.5·16-s + 1.23e3·17-s + 745.·18-s + 2.14e3·19-s − 1.75e3·21-s + 1.11e3·22-s + 2.27e3·23-s + 1.72e3·24-s − 4.20e3·26-s + 729·27-s − 1.03e4·28-s + 6.56e3·29-s + 6.34e3·31-s − 5.44e3·32-s + 1.08e3·33-s + 1.13e4·34-s + ⋯ |
| L(s) = 1 | + 1.62·2-s + 0.577·3-s + 1.65·4-s + 0.939·6-s − 1.50·7-s + 1.05·8-s + 0.333·9-s + 0.301·11-s + 0.952·12-s − 0.750·13-s − 2.45·14-s + 0.0727·16-s + 1.03·17-s + 0.542·18-s + 1.36·19-s − 0.869·21-s + 0.490·22-s + 0.895·23-s + 0.611·24-s − 1.22·26-s + 0.192·27-s − 2.48·28-s + 1.44·29-s + 1.18·31-s − 0.939·32-s + 0.174·33-s + 1.69·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.946331916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.946331916\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 - 9.20T + 32T^{2} \) |
| 7 | \( 1 + 195.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 457.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.23e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.14e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.27e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.56e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.34e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.37e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.97e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.90e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.26e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.05e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.35e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.90e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.47e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 8.96e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.33e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.78e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.68e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.17e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.56e5T + 8.58e9T^{2} \) |
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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
−9.731431887167850381453038533188, −8.639717165283244653014041277789, −7.29217038294061981322305968784, −6.78371537055980418124855664778, −5.82022252530658830829774920649, −4.97732614396617454141738492569, −3.91438715841361645112203213419, −3.08047207459736553568818801655, −2.65967792702599822114412530642, −0.915380392256430909111065632050,
0.915380392256430909111065632050, 2.65967792702599822114412530642, 3.08047207459736553568818801655, 3.91438715841361645112203213419, 4.97732614396617454141738492569, 5.82022252530658830829774920649, 6.78371537055980418124855664778, 7.29217038294061981322305968784, 8.639717165283244653014041277789, 9.731431887167850381453038533188
