| L(s) = 1 | − 5.91·2-s − 93.0·4-s − 125·5-s − 1.49e3·7-s + 1.30e3·8-s + 738.·10-s − 3.57e3·11-s − 109.·13-s + 8.82e3·14-s + 4.18e3·16-s − 2.44e4·17-s + 2.84e4·19-s + 1.16e4·20-s + 2.11e4·22-s + 5.40e4·23-s + 1.56e4·25-s + 648.·26-s + 1.38e5·28-s − 1.01e5·29-s + 1.01e4·31-s − 1.92e5·32-s + 1.44e5·34-s + 1.86e5·35-s + 5.72e5·37-s − 1.68e5·38-s − 1.63e5·40-s + 2.90e5·41-s + ⋯ |
| L(s) = 1 | − 0.522·2-s − 0.727·4-s − 0.447·5-s − 1.64·7-s + 0.902·8-s + 0.233·10-s − 0.810·11-s − 0.0138·13-s + 0.859·14-s + 0.255·16-s − 1.20·17-s + 0.953·19-s + 0.325·20-s + 0.423·22-s + 0.925·23-s + 0.199·25-s + 0.00723·26-s + 1.19·28-s − 0.771·29-s + 0.0611·31-s − 1.03·32-s + 0.630·34-s + 0.735·35-s + 1.85·37-s − 0.498·38-s − 0.403·40-s + 0.659·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 125T \) |
| good | 2 | \( 1 + 5.91T + 128T^{2} \) |
| 7 | \( 1 + 1.49e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.57e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 109.T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.44e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.84e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.40e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.01e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.01e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.72e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.90e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.37e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 8.71e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 3.54e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.00e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.54e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.04e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.07e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.39e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 8.42e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.91e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.87e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 3.31e6T + 8.07e13T^{2} \) |
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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
−9.456418272772226172330772726724, −8.973445371795620191559232999272, −7.80000440143565807864751371630, −7.02585927250443752108804160076, −5.85265529158592250292406568497, −4.69125214376494952025273002803, −3.64205310116475469572270518831, −2.62364769129645865825386974417, −0.806001948371993632433017301467, 0,
0.806001948371993632433017301467, 2.62364769129645865825386974417, 3.64205310116475469572270518831, 4.69125214376494952025273002803, 5.85265529158592250292406568497, 7.02585927250443752108804160076, 7.80000440143565807864751371630, 8.973445371795620191559232999272, 9.456418272772226172330772726724
