| L(s) = 1 | + 17.4·2-s + 175.·4-s + 74.3·5-s + 818.·8-s + 1.29e3·10-s + 1.95e3·11-s + 3.40e3·13-s − 8.15e3·16-s − 2.23e4·17-s − 4.98e4·19-s + 1.30e4·20-s + 3.40e4·22-s + 1.19e4·23-s − 7.25e4·25-s + 5.93e4·26-s − 1.62e5·29-s − 3.16e4·31-s − 2.46e5·32-s − 3.88e5·34-s + 3.74e5·37-s − 8.66e5·38-s + 6.08e4·40-s + 4.52e5·41-s + 6.52e5·43-s + 3.42e5·44-s + 2.08e5·46-s − 9.19e5·47-s + ⋯ |
| L(s) = 1 | + 1.53·2-s + 1.36·4-s + 0.266·5-s + 0.564·8-s + 0.409·10-s + 0.443·11-s + 0.430·13-s − 0.497·16-s − 1.10·17-s − 1.66·19-s + 0.363·20-s + 0.682·22-s + 0.205·23-s − 0.929·25-s + 0.661·26-s − 1.24·29-s − 0.190·31-s − 1.33·32-s − 1.69·34-s + 1.21·37-s − 2.56·38-s + 0.150·40-s + 1.02·41-s + 1.25·43-s + 0.606·44-s + 0.315·46-s − 1.29·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 17.4T + 128T^{2} \) |
| 5 | \( 1 - 74.3T + 7.81e4T^{2} \) |
| 11 | \( 1 - 1.95e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 3.40e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.23e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.98e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.19e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.62e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 3.16e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.74e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.52e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.52e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.19e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.54e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.20e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.78e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.08e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.09e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.17e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 8.16e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.71e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.41e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 6.18e6T + 8.07e13T^{2} \) |
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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
−9.481374482258651645430034852499, −8.643734722921897277454275415903, −7.29327871976403605540182138822, −6.25800009309190004627540825722, −5.80084227006976851980937875844, −4.43674203308665382019972889800, −3.99927391620868351419962432231, −2.68155625679211959998558455908, −1.77426562788934346177139543319, 0,
1.77426562788934346177139543319, 2.68155625679211959998558455908, 3.99927391620868351419962432231, 4.43674203308665382019972889800, 5.80084227006976851980937875844, 6.25800009309190004627540825722, 7.29327871976403605540182138822, 8.643734722921897277454275415903, 9.481374482258651645430034852499
