| L(s) = 1 | + 2.01e5·3-s − 2.11e7·5-s + 7.32e8·7-s + 3.02e10·9-s − 5.59e9·11-s − 6.30e10·13-s − 4.26e12·15-s + 1.35e13·17-s + 1.39e13·19-s + 1.47e14·21-s − 2.80e14·23-s − 2.92e13·25-s + 3.99e15·27-s + 1.19e15·29-s + 3.88e15·31-s − 1.12e15·33-s − 1.55e16·35-s + 2.72e16·37-s − 1.27e16·39-s − 6.89e16·41-s + 3.24e16·43-s − 6.40e17·45-s − 2.07e17·47-s − 2.12e16·49-s + 2.73e18·51-s − 6.38e17·53-s + 1.18e17·55-s + ⋯ |
| L(s) = 1 | + 1.97·3-s − 0.968·5-s + 0.980·7-s + 2.89·9-s − 0.0650·11-s − 0.126·13-s − 1.91·15-s + 1.63·17-s + 0.520·19-s + 1.93·21-s − 1.41·23-s − 0.0614·25-s + 3.73·27-s + 0.528·29-s + 0.851·31-s − 0.128·33-s − 0.950·35-s + 0.929·37-s − 0.250·39-s − 0.802·41-s + 0.229·43-s − 2.80·45-s − 0.575·47-s − 0.0381·49-s + 3.21·51-s − 0.501·53-s + 0.0630·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(3.807723439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.807723439\) |
| \(L(\frac{23}{2})\) |
|
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 | \( 1 - 2.01e5T + 1.04e10T^{2} \) |
| 5 | \( 1 + 2.11e7T + 4.76e14T^{2} \) |
| 7 | \( 1 - 7.32e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 5.59e9T + 7.40e21T^{2} \) |
| 13 | \( 1 + 6.30e10T + 2.47e23T^{2} \) |
| 17 | \( 1 - 1.35e13T + 6.90e25T^{2} \) |
| 19 | \( 1 - 1.39e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 2.80e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 1.19e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 3.88e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 2.72e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 6.89e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 3.24e16T + 2.00e34T^{2} \) |
| 47 | \( 1 + 2.07e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 6.38e17T + 1.62e36T^{2} \) |
| 59 | \( 1 + 3.04e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 5.64e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 3.96e18T + 2.22e38T^{2} \) |
| 71 | \( 1 + 2.61e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 1.37e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 1.18e20T + 7.08e39T^{2} \) |
| 83 | \( 1 + 1.60e19T + 1.99e40T^{2} \) |
| 89 | \( 1 + 2.30e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 2.72e20T + 5.27e41T^{2} \) |
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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
−15.91990633820001058089742734909, −14.78307138738395676748998759020, −13.90379485449520596106686031128, −12.09824568880882032340881025968, −9.895479286407433792145473133090, −8.190056392350706659114022960019, −7.68410572211849094473054260583, −4.33169058985607964794153555909, −3.08096260778779607030523426577, −1.46447986862364449978882829381,
1.46447986862364449978882829381, 3.08096260778779607030523426577, 4.33169058985607964794153555909, 7.68410572211849094473054260583, 8.190056392350706659114022960019, 9.895479286407433792145473133090, 12.09824568880882032340881025968, 13.90379485449520596106686031128, 14.78307138738395676748998759020, 15.91990633820001058089742734909
