| L(s) = 1 | + 243·3-s − 3.47e3·5-s − 1.68e4·7-s + 5.90e4·9-s + 4.70e5·11-s − 8.80e5·13-s − 8.43e5·15-s − 9.10e6·17-s + 9.11e6·19-s − 4.08e6·21-s + 4.77e7·23-s − 3.67e7·25-s + 1.43e7·27-s + 1.25e8·29-s − 6.67e7·31-s + 1.14e8·33-s + 5.83e7·35-s + 2.26e8·37-s − 2.14e8·39-s − 3.82e8·41-s + 1.67e9·43-s − 2.05e8·45-s − 5.37e8·47-s + 2.82e8·49-s − 2.21e9·51-s + 9.26e8·53-s − 1.63e9·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.497·5-s − 0.377·7-s + 0.333·9-s + 0.881·11-s − 0.658·13-s − 0.286·15-s − 1.55·17-s + 0.844·19-s − 0.218·21-s + 1.54·23-s − 0.752·25-s + 0.192·27-s + 1.13·29-s − 0.418·31-s + 0.508·33-s + 0.187·35-s + 0.536·37-s − 0.379·39-s − 0.515·41-s + 1.73·43-s − 0.165·45-s − 0.341·47-s + 0.142·49-s − 0.897·51-s + 0.304·53-s − 0.437·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.234468501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.234468501\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 243T \) |
| 7 | \( 1 + 1.68e4T \) |
| good | 5 | \( 1 + 3.47e3T + 4.88e7T^{2} \) |
| 11 | \( 1 - 4.70e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 8.80e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 9.10e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 9.11e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 4.77e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.25e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 6.67e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 2.26e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 3.82e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.67e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 5.37e8T + 2.47e18T^{2} \) |
| 53 | \( 1 - 9.26e8T + 9.26e18T^{2} \) |
| 59 | \( 1 - 3.01e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 3.97e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.67e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.25e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 3.68e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 4.42e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 6.38e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 3.65e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.18e11T + 7.15e21T^{2} \) |
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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
−12.04684595132759101491431649309, −11.01206217207425412077317335079, −9.586535423348097162663006554506, −8.818859834273524170299306901653, −7.48790537910218017782620232946, −6.53864460257471746967412696671, −4.76698473921346425064969489995, −3.61390235454927954414086754652, −2.37060798408507214486409905985, −0.77299785247536329212293091616,
0.77299785247536329212293091616, 2.37060798408507214486409905985, 3.61390235454927954414086754652, 4.76698473921346425064969489995, 6.53864460257471746967412696671, 7.48790537910218017782620232946, 8.818859834273524170299306901653, 9.586535423348097162663006554506, 11.01206217207425412077317335079, 12.04684595132759101491431649309
