| L(s) = 1 | − 5.31e5·3-s + 9.25e8·5-s + 2.69e10·7-s + 2.82e11·9-s + 1.97e13·11-s − 1.26e14·13-s − 4.91e14·15-s + 6.34e14·17-s + 6.86e15·19-s − 1.43e16·21-s + 1.83e17·23-s + 5.58e17·25-s − 1.50e17·27-s + 1.44e17·29-s − 2.05e18·31-s − 1.04e19·33-s + 2.49e19·35-s + 7.76e19·37-s + 6.72e19·39-s − 4.12e19·41-s + 3.81e20·43-s + 2.61e20·45-s + 3.11e20·47-s − 6.12e20·49-s − 3.37e20·51-s − 4.51e21·53-s + 1.82e22·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.69·5-s + 0.737·7-s + 0.333·9-s + 1.89·11-s − 1.50·13-s − 0.978·15-s + 0.264·17-s + 0.711·19-s − 0.425·21-s + 1.74·23-s + 1.87·25-s − 0.192·27-s + 0.0758·29-s − 0.469·31-s − 1.09·33-s + 1.24·35-s + 1.93·37-s + 0.870·39-s − 0.285·41-s + 1.45·43-s + 0.565·45-s + 0.391·47-s − 0.456·49-s − 0.152·51-s − 1.26·53-s + 3.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(3.895720289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.895720289\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 5.31e5T \) |
| good | 5 | \( 1 - 9.25e8T + 2.98e17T^{2} \) |
| 7 | \( 1 - 2.69e10T + 1.34e21T^{2} \) |
| 11 | \( 1 - 1.97e13T + 1.08e26T^{2} \) |
| 13 | \( 1 + 1.26e14T + 7.05e27T^{2} \) |
| 17 | \( 1 - 6.34e14T + 5.77e30T^{2} \) |
| 19 | \( 1 - 6.86e15T + 9.30e31T^{2} \) |
| 23 | \( 1 - 1.83e17T + 1.10e34T^{2} \) |
| 29 | \( 1 - 1.44e17T + 3.63e36T^{2} \) |
| 31 | \( 1 + 2.05e18T + 1.92e37T^{2} \) |
| 37 | \( 1 - 7.76e19T + 1.60e39T^{2} \) |
| 41 | \( 1 + 4.12e19T + 2.08e40T^{2} \) |
| 43 | \( 1 - 3.81e20T + 6.86e40T^{2} \) |
| 47 | \( 1 - 3.11e20T + 6.34e41T^{2} \) |
| 53 | \( 1 + 4.51e21T + 1.27e43T^{2} \) |
| 59 | \( 1 + 4.18e21T + 1.86e44T^{2} \) |
| 61 | \( 1 - 8.65e21T + 4.29e44T^{2} \) |
| 67 | \( 1 + 4.85e22T + 4.48e45T^{2} \) |
| 71 | \( 1 - 2.41e21T + 1.91e46T^{2} \) |
| 73 | \( 1 + 1.69e23T + 3.82e46T^{2} \) |
| 79 | \( 1 + 6.93e22T + 2.75e47T^{2} \) |
| 83 | \( 1 + 8.18e23T + 9.48e47T^{2} \) |
| 89 | \( 1 - 2.26e24T + 5.42e48T^{2} \) |
| 97 | \( 1 - 1.64e24T + 4.66e49T^{2} \) |
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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
−10.95292776912770814104895987525, −9.636483199295172000323367444081, −9.214376839156847695506917454413, −7.32894347760283771758023161680, −6.35340277137782195436232473320, −5.37591136679692870449459774969, −4.51205024996316187400157104734, −2.74639370433298789963973781537, −1.57903702028374986519755562777, −0.971171860366175599647786922321,
0.971171860366175599647786922321, 1.57903702028374986519755562777, 2.74639370433298789963973781537, 4.51205024996316187400157104734, 5.37591136679692870449459774969, 6.35340277137782195436232473320, 7.32894347760283771758023161680, 9.214376839156847695506917454413, 9.636483199295172000323367444081, 10.95292776912770814104895987525
