| L(s) = 1 | − 708.·2-s + 1.98e5·3-s − 7.88e6·4-s + 3.63e7·5-s − 1.40e8·6-s + 1.15e10·8-s − 5.48e10·9-s − 2.57e10·10-s − 2.09e11·11-s − 1.56e12·12-s − 1.80e12·13-s + 7.21e12·15-s + 5.79e13·16-s + 6.17e13·17-s + 3.88e13·18-s + 3.57e14·19-s − 2.86e14·20-s + 1.48e14·22-s + 2.91e14·23-s + 2.28e15·24-s − 1.05e16·25-s + 1.27e15·26-s − 2.95e16·27-s + 6.50e16·29-s − 5.10e15·30-s + 1.04e17·31-s − 1.37e17·32-s + ⋯ |
| L(s) = 1 | − 0.244·2-s + 0.646·3-s − 0.940·4-s + 0.333·5-s − 0.158·6-s + 0.474·8-s − 0.582·9-s − 0.0814·10-s − 0.221·11-s − 0.607·12-s − 0.278·13-s + 0.215·15-s + 0.824·16-s + 0.437·17-s + 0.142·18-s + 0.703·19-s − 0.313·20-s + 0.0540·22-s + 0.0638·23-s + 0.306·24-s − 0.889·25-s + 0.0682·26-s − 1.02·27-s + 0.989·29-s − 0.0526·30-s + 0.735·31-s − 0.676·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + 708.T + 8.38e6T^{2} \) |
| 3 | \( 1 - 1.98e5T + 9.41e10T^{2} \) |
| 5 | \( 1 - 3.63e7T + 1.19e16T^{2} \) |
| 11 | \( 1 + 2.09e11T + 8.95e23T^{2} \) |
| 13 | \( 1 + 1.80e12T + 4.17e25T^{2} \) |
| 17 | \( 1 - 6.17e13T + 1.99e28T^{2} \) |
| 19 | \( 1 - 3.57e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 2.91e14T + 2.08e31T^{2} \) |
| 29 | \( 1 - 6.50e16T + 4.31e33T^{2} \) |
| 31 | \( 1 - 1.04e17T + 2.00e34T^{2} \) |
| 37 | \( 1 + 4.18e17T + 1.17e36T^{2} \) |
| 41 | \( 1 - 6.10e18T + 1.24e37T^{2} \) |
| 43 | \( 1 + 6.48e18T + 3.71e37T^{2} \) |
| 47 | \( 1 - 7.19e18T + 2.87e38T^{2} \) |
| 53 | \( 1 + 1.00e20T + 4.55e39T^{2} \) |
| 59 | \( 1 - 3.01e20T + 5.36e40T^{2} \) |
| 61 | \( 1 - 4.76e19T + 1.15e41T^{2} \) |
| 67 | \( 1 - 8.72e20T + 9.99e41T^{2} \) |
| 71 | \( 1 + 3.00e21T + 3.79e42T^{2} \) |
| 73 | \( 1 + 4.53e20T + 7.18e42T^{2} \) |
| 79 | \( 1 + 6.15e21T + 4.42e43T^{2} \) |
| 83 | \( 1 + 6.71e21T + 1.37e44T^{2} \) |
| 89 | \( 1 - 3.62e22T + 6.85e44T^{2} \) |
| 97 | \( 1 - 1.11e23T + 4.96e45T^{2} \) |
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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
−10.21189390649461994794338415828, −9.418572287681956284521159568260, −8.451229833832580824424342120550, −7.61109731862684668310774447788, −5.90484547360759199634772379406, −4.85663331170771369271818716235, −3.57958321804786011041845471517, −2.52623558857802003932298096608, −1.15479283817962763496051721819, 0,
1.15479283817962763496051721819, 2.52623558857802003932298096608, 3.57958321804786011041845471517, 4.85663331170771369271818716235, 5.90484547360759199634772379406, 7.61109731862684668310774447788, 8.451229833832580824424342120550, 9.418572287681956284521159568260, 10.21189390649461994794338415828
