| L(s) = 1 | + 1.22e6·2-s − 7.30e12·4-s + 8.84e14·5-s + 1.48e18·7-s − 1.96e19·8-s + 1.07e21·10-s − 1.46e21·11-s − 4.36e23·13-s + 1.80e24·14-s + 4.02e25·16-s − 3.51e26·17-s − 1.50e27·19-s − 6.45e27·20-s − 1.78e27·22-s + 1.10e29·23-s − 3.55e29·25-s − 5.33e29·26-s − 1.08e31·28-s + 3.00e31·29-s + 1.83e32·31-s + 2.22e32·32-s − 4.29e32·34-s + 1.30e33·35-s − 9.66e33·37-s − 1.83e33·38-s − 1.73e34·40-s + 5.73e34·41-s + ⋯ |
| L(s) = 1 | + 0.411·2-s − 0.830·4-s + 0.829·5-s + 1.00·7-s − 0.753·8-s + 0.341·10-s − 0.0596·11-s − 0.489·13-s + 0.412·14-s + 0.519·16-s − 1.23·17-s − 0.481·19-s − 0.688·20-s − 0.0245·22-s + 0.582·23-s − 0.312·25-s − 0.201·26-s − 0.831·28-s + 1.08·29-s + 1.58·31-s + 0.967·32-s − 0.508·34-s + 0.830·35-s − 1.85·37-s − 0.198·38-s − 0.625·40-s + 1.21·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(44-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+43/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(22)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{45}{2})\) |
|
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 | \( 1 - 1.22e6T + 8.79e12T^{2} \) |
| 5 | \( 1 - 8.84e14T + 1.13e30T^{2} \) |
| 7 | \( 1 - 1.48e18T + 2.18e36T^{2} \) |
| 11 | \( 1 + 1.46e21T + 6.02e44T^{2} \) |
| 13 | \( 1 + 4.36e23T + 7.93e47T^{2} \) |
| 17 | \( 1 + 3.51e26T + 8.11e52T^{2} \) |
| 19 | \( 1 + 1.50e27T + 9.69e54T^{2} \) |
| 23 | \( 1 - 1.10e29T + 3.58e58T^{2} \) |
| 29 | \( 1 - 3.00e31T + 7.64e62T^{2} \) |
| 31 | \( 1 - 1.83e32T + 1.34e64T^{2} \) |
| 37 | \( 1 + 9.66e33T + 2.70e67T^{2} \) |
| 41 | \( 1 - 5.73e34T + 2.23e69T^{2} \) |
| 43 | \( 1 + 1.24e35T + 1.73e70T^{2} \) |
| 47 | \( 1 - 6.52e35T + 7.94e71T^{2} \) |
| 53 | \( 1 + 2.03e37T + 1.39e74T^{2} \) |
| 59 | \( 1 - 7.05e37T + 1.40e76T^{2} \) |
| 61 | \( 1 + 4.16e38T + 5.87e76T^{2} \) |
| 67 | \( 1 + 1.46e39T + 3.32e78T^{2} \) |
| 71 | \( 1 + 6.58e39T + 4.01e79T^{2} \) |
| 73 | \( 1 - 8.76e39T + 1.32e80T^{2} \) |
| 79 | \( 1 + 3.63e39T + 3.96e81T^{2} \) |
| 83 | \( 1 - 2.43e41T + 3.31e82T^{2} \) |
| 89 | \( 1 + 6.12e41T + 6.66e83T^{2} \) |
| 97 | \( 1 - 8.54e41T + 2.69e85T^{2} \) |
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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
−12.10419273732563174805415750941, −10.56099030933014319195793060357, −9.258530204666739095005567942067, −8.207942400814448604579588527013, −6.43192025714996981031852782694, −5.11200289531428934062618680968, −4.38929632942187844921073508262, −2.69045822074549253660302622043, −1.44878762555652497073851291603, 0,
1.44878762555652497073851291603, 2.69045822074549253660302622043, 4.38929632942187844921073508262, 5.11200289531428934062618680968, 6.43192025714996981031852782694, 8.207942400814448604579588527013, 9.258530204666739095005567942067, 10.56099030933014319195793060357, 12.10419273732563174805415750941
