| L(s) = 1 | − 81·3-s − 287.·5-s − 6.23e3·7-s + 6.56e3·9-s − 1.46e4·11-s − 1.63e5·13-s + 2.33e4·15-s + 5.31e5·17-s − 8.94e5·19-s + 5.05e5·21-s − 1.04e6·23-s − 1.87e6·25-s − 5.31e5·27-s + 1.62e6·29-s + 2.00e6·31-s + 1.18e6·33-s + 1.79e6·35-s + 3.24e6·37-s + 1.32e7·39-s − 2.98e7·41-s − 2.52e7·43-s − 1.88e6·45-s + 8.29e6·47-s − 1.48e6·49-s − 4.30e7·51-s + 5.36e7·53-s + 4.21e6·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.206·5-s − 0.981·7-s + 0.333·9-s − 0.301·11-s − 1.59·13-s + 0.118·15-s + 1.54·17-s − 1.57·19-s + 0.566·21-s − 0.781·23-s − 0.957·25-s − 0.192·27-s + 0.426·29-s + 0.390·31-s + 0.174·33-s + 0.202·35-s + 0.284·37-s + 0.919·39-s − 1.64·41-s − 1.12·43-s − 0.0686·45-s + 0.247·47-s − 0.0367·49-s − 0.891·51-s + 0.934·53-s + 0.0621·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.6295302457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6295302457\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 81T \) |
| 11 | \( 1 + 1.46e4T \) |
| good | 5 | \( 1 + 287.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 6.23e3T + 4.03e7T^{2} \) |
| 13 | \( 1 + 1.63e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 5.31e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 8.94e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.04e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.62e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.00e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 3.24e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.98e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.52e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 8.29e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.36e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.55e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.41e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.92e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.49e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 4.31e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.96e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.56e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.92e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.91e8T + 7.60e17T^{2} \) |
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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
−11.79443984218737100766284417716, −10.15110482749973228533199271877, −9.935172968374666513062944860465, −8.260935494068135534398028026389, −7.14021358692210235873535581339, −6.10019764838724164795085287185, −4.95514920229678738404441237371, −3.61676045096822425312745281381, −2.21695504938107368042889941763, −0.40554097262556698066990542453,
0.40554097262556698066990542453, 2.21695504938107368042889941763, 3.61676045096822425312745281381, 4.95514920229678738404441237371, 6.10019764838724164795085287185, 7.14021358692210235873535581339, 8.260935494068135534398028026389, 9.935172968374666513062944860465, 10.15110482749973228533199271877, 11.79443984218737100766284417716
