| L(s) = 1 | − 83.7·2-s + 503.·3-s + 4.96e3·4-s + 3.12e3·5-s − 4.21e4·6-s − 2.44e5·8-s + 7.60e4·9-s − 2.61e5·10-s + 3.39e5·11-s + 2.49e6·12-s − 2.02e6·13-s + 1.57e6·15-s + 1.02e7·16-s + 2.45e6·17-s − 6.37e6·18-s + 4.08e6·19-s + 1.55e7·20-s − 2.84e7·22-s + 2.86e7·23-s − 1.22e8·24-s + 9.76e6·25-s + 1.69e8·26-s − 5.08e7·27-s − 9.41e6·29-s − 1.31e8·30-s − 2.99e8·31-s − 3.60e8·32-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 1.19·3-s + 2.42·4-s + 0.447·5-s − 2.21·6-s − 2.63·8-s + 0.429·9-s − 0.827·10-s + 0.636·11-s + 2.89·12-s − 1.51·13-s + 0.534·15-s + 2.44·16-s + 0.418·17-s − 0.794·18-s + 0.378·19-s + 1.08·20-s − 1.17·22-s + 0.928·23-s − 3.14·24-s + 0.199·25-s + 2.79·26-s − 0.682·27-s − 0.0852·29-s − 0.989·30-s − 1.87·31-s − 1.89·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.443477366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.443477366\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 3.12e3T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 83.7T + 2.04e3T^{2} \) |
| 3 | \( 1 - 503.T + 1.77e5T^{2} \) |
| 11 | \( 1 - 3.39e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 2.02e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 2.45e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 4.08e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 2.86e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 9.41e6T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.99e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 4.57e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.83e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 6.56e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.97e8T + 2.47e18T^{2} \) |
| 53 | \( 1 - 5.15e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 6.62e8T + 3.01e19T^{2} \) |
| 61 | \( 1 + 5.58e8T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.01e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.78e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.33e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 1.24e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 3.37e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 2.94e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.13e11T + 7.15e21T^{2} \) |
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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
−9.649524967431193453744246113398, −9.256130100963732513939868466190, −8.487631451250251431369362547059, −7.46355850028373552754832820108, −6.95693002571773374783413553515, −5.43734713114021171142596544216, −3.46506680092614851900983512190, −2.44620412736602135786798126530, −1.79021339261273909024440086227, −0.62196683767690785674218394785,
0.62196683767690785674218394785, 1.79021339261273909024440086227, 2.44620412736602135786798126530, 3.46506680092614851900983512190, 5.43734713114021171142596544216, 6.95693002571773374783413553515, 7.46355850028373552754832820108, 8.487631451250251431369362547059, 9.256130100963732513939868466190, 9.649524967431193453744246113398
