| L(s) = 1 | + 11.3i·2-s − 128.·4-s + 1.08e3i·5-s + 907.·7-s − 1.44e3i·8-s − 1.22e4·10-s − 2.19e4i·11-s + 4.47e4·13-s + 1.02e4i·14-s + 1.63e4·16-s − 1.37e5i·17-s + 9.56e4·19-s − 1.38e5i·20-s + 2.48e5·22-s − 1.05e5i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s + 1.73i·5-s + 0.377·7-s − 0.353i·8-s − 1.22·10-s − 1.49i·11-s + 1.56·13-s + 0.267i·14-s + 0.250·16-s − 1.65i·17-s + 0.733·19-s − 0.865i·20-s + 1.05·22-s − 0.378i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.111101224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.111101224\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 11.3iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 907.T \) |
| good | 5 | \( 1 - 1.08e3iT - 3.90e5T^{2} \) |
| 11 | \( 1 + 2.19e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 4.47e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 1.37e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 9.56e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + 1.05e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 7.75e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 6.31e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 3.05e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 7.33e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 1.43e5T + 1.16e13T^{2} \) |
| 47 | \( 1 + 8.56e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 6.76e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 1.41e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.46e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 1.87e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 3.91e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.96e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 4.18e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 7.47e6iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 8.86e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 8.12e7T + 7.83e15T^{2} \) |
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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
−11.40798333995337852371057048721, −11.13798273620719225778713655449, −9.811636735845241474526552303930, −8.511283979047807448898444047476, −7.47507756023770475020203912260, −6.45452360059013689259571191257, −5.62856990649252306331647515780, −3.76588921177263678849769901933, −2.77889466161049540571916440408, −0.67241009438293292160598101026,
1.13969096329934995023814381881, 1.70729483086123103900336794033, 3.84352446635358409142936088508, 4.71065152754006712529141467109, 5.83796388676675145088621418216, 7.80105061068563481573654802086, 8.729027801683241387492201096362, 9.519696585750943631373791663363, 10.74702487882712552444956332850, 11.87108596885249690340788108406
