| L(s) = 1 | + 52.9·2-s + 755.·4-s + 9.07e3·5-s + 9.37e3·7-s − 6.84e4·8-s + 4.80e5·10-s + 1.38e5·11-s + 6.94e4·13-s + 4.96e5·14-s − 5.17e6·16-s + 8.48e6·17-s + 9.50e6·19-s + 6.86e6·20-s + 7.33e6·22-s + 3.59e7·23-s + 3.35e7·25-s + 3.67e6·26-s + 7.08e6·28-s − 3.94e7·29-s − 2.31e8·31-s − 1.33e8·32-s + 4.49e8·34-s + 8.51e7·35-s − 2.47e8·37-s + 5.03e8·38-s − 6.21e8·40-s + 1.32e9·41-s + ⋯ |
| L(s) = 1 | + 1.17·2-s + 0.368·4-s + 1.29·5-s + 0.210·7-s − 0.738·8-s + 1.52·10-s + 0.259·11-s + 0.0518·13-s + 0.246·14-s − 1.23·16-s + 1.44·17-s + 0.880·19-s + 0.479·20-s + 0.303·22-s + 1.16·23-s + 0.688·25-s + 0.0607·26-s + 0.0778·28-s − 0.357·29-s − 1.45·31-s − 0.704·32-s + 1.69·34-s + 0.273·35-s − 0.586·37-s + 1.03·38-s − 0.959·40-s + 1.78·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(5.165282229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.165282229\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 52.9T + 2.04e3T^{2} \) |
| 5 | \( 1 - 9.07e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 9.37e3T + 1.97e9T^{2} \) |
| 11 | \( 1 - 1.38e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 6.94e4T + 1.79e12T^{2} \) |
| 17 | \( 1 - 8.48e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 9.50e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 3.59e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 3.94e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.31e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 2.47e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.32e9T + 5.50e17T^{2} \) |
| 43 | \( 1 - 7.82e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.08e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 2.90e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 3.55e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 1.23e10T + 4.35e19T^{2} \) |
| 67 | \( 1 - 2.10e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.67e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.30e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 3.59e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 1.62e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 8.11e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 7.78e10T + 7.15e21T^{2} \) |
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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
−12.48037143101721623656254592075, −11.25336776372854731878794446791, −9.844752183853048116036256360374, −8.993639726993342058321552217956, −7.19788303973122626245859572550, −5.74706703022550194691466926835, −5.31853559914111977527011611004, −3.76749561626760771624520900090, −2.54968451536481786952512869678, −1.11000239915893148657753660411,
1.11000239915893148657753660411, 2.54968451536481786952512869678, 3.76749561626760771624520900090, 5.31853559914111977527011611004, 5.74706703022550194691466926835, 7.19788303973122626245859572550, 8.993639726993342058321552217956, 9.844752183853048116036256360374, 11.25336776372854731878794446791, 12.48037143101721623656254592075
