| L(s) = 1 | + 16·2-s − 144.·3-s + 256·4-s − 1.70e3·5-s − 2.31e3·6-s + 8.66e3·7-s + 4.09e3·8-s + 1.24e3·9-s − 2.73e4·10-s − 3.70e4·12-s − 6.78e4·13-s + 1.38e5·14-s + 2.46e5·15-s + 6.55e4·16-s − 2.49e5·17-s + 1.99e4·18-s + 2.93e5·19-s − 4.36e5·20-s − 1.25e6·21-s + 1.10e6·23-s − 5.92e5·24-s + 9.59e5·25-s − 1.08e6·26-s + 2.66e6·27-s + 2.21e6·28-s + 5.23e6·29-s + 3.95e6·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.03·3-s + 0.5·4-s − 1.22·5-s − 0.729·6-s + 1.36·7-s + 0.353·8-s + 0.0633·9-s − 0.863·10-s − 0.515·12-s − 0.658·13-s + 0.964·14-s + 1.25·15-s + 0.250·16-s − 0.724·17-s + 0.0448·18-s + 0.516·19-s − 0.610·20-s − 1.40·21-s + 0.821·23-s − 0.364·24-s + 0.491·25-s − 0.465·26-s + 0.965·27-s + 0.681·28-s + 1.37·29-s + 0.890·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 16T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 144.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.70e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 8.66e3T + 4.03e7T^{2} \) |
| 13 | \( 1 + 6.78e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.49e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.93e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.10e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.23e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.38e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 7.34e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.01e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.56e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.88e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 8.68e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.82e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.22e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 6.80e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.67e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.51e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.95e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.61e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.63e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.06e8T + 7.60e17T^{2} \) |
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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
−10.72615187005007932219295154230, −8.874285172896023432802959542757, −7.75352349947900540972507138810, −7.04745321725954955542500267968, −5.69336848298982440298722123794, −4.83730443843404738935053620995, −4.18617761333048992545644152489, −2.69972060068398363807437585177, −1.17272333939824782661214309302, 0,
1.17272333939824782661214309302, 2.69972060068398363807437585177, 4.18617761333048992545644152489, 4.83730443843404738935053620995, 5.69336848298982440298722123794, 7.04745321725954955542500267968, 7.75352349947900540972507138810, 8.874285172896023432802959542757, 10.72615187005007932219295154230
