| L(s) = 1 | − 219.·3-s + 4.58e3·5-s + 2.52e4·7-s − 1.28e5·9-s + 5.59e5·11-s + 2.42e6·13-s − 1.00e6·15-s + 1.46e6·17-s − 1.30e7·19-s − 5.54e6·21-s + 2.87e7·23-s − 2.77e7·25-s + 6.72e7·27-s + 7.38e7·29-s + 1.68e8·31-s − 1.22e8·33-s + 1.15e8·35-s − 2.80e8·37-s − 5.33e8·39-s − 7.46e7·41-s − 1.18e9·43-s − 5.91e8·45-s + 1.22e9·47-s − 1.33e9·49-s − 3.22e8·51-s − 1.31e9·53-s + 2.56e9·55-s + ⋯ |
| L(s) = 1 | − 0.521·3-s + 0.656·5-s + 0.568·7-s − 0.727·9-s + 1.04·11-s + 1.81·13-s − 0.342·15-s + 0.250·17-s − 1.20·19-s − 0.296·21-s + 0.931·23-s − 0.568·25-s + 0.901·27-s + 0.668·29-s + 1.05·31-s − 0.546·33-s + 0.373·35-s − 0.664·37-s − 0.946·39-s − 0.100·41-s − 1.23·43-s − 0.477·45-s + 0.782·47-s − 0.677·49-s − 0.130·51-s − 0.432·53-s + 0.687·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.500417829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.500417829\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 219.T + 1.77e5T^{2} \) |
| 5 | \( 1 - 4.58e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 2.52e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 5.59e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 2.42e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 1.46e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.30e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 2.87e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 7.38e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.68e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 2.80e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 7.46e7T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.18e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.22e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.31e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 1.82e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 7.20e8T + 4.35e19T^{2} \) |
| 67 | \( 1 - 5.93e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 4.84e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 3.22e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 3.97e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 3.24e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 9.37e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.03e11T + 7.15e21T^{2} \) |
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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
−11.22620618925976251144716354347, −10.40849571321061662977571022484, −8.977321981553810359858437946824, −8.322046208946408835465979157021, −6.51684417956069801660576529205, −5.99490601993594366420110379408, −4.72477064063293854561073326865, −3.38964356164787897231755176995, −1.80826416042229734104220207167, −0.834036260271529559838181201490,
0.834036260271529559838181201490, 1.80826416042229734104220207167, 3.38964356164787897231755176995, 4.72477064063293854561073326865, 5.99490601993594366420110379408, 6.51684417956069801660576529205, 8.322046208946408835465979157021, 8.977321981553810359858437946824, 10.40849571321061662977571022484, 11.22620618925976251144716354347
