L(s) = 1 | + 173. i·5-s − 484·7-s + 1.34e3i·11-s − 3.36e3·13-s − 12.7i·17-s − 5.74e3·19-s + 3.37e3i·23-s − 1.46e4·25-s + 2.93e4i·29-s − 3.97e4·31-s − 8.41e4i·35-s − 5.25e4·37-s + 3.70e4i·41-s − 3.80e3·43-s − 7.67e4i·47-s + ⋯ |
L(s) = 1 | + 1.39i·5-s − 1.41·7-s + 1.00i·11-s − 1.53·13-s − 0.00259i·17-s − 0.837·19-s + 0.277i·23-s − 0.936·25-s + 1.20i·29-s − 1.33·31-s − 1.96i·35-s − 1.03·37-s + 0.537i·41-s − 0.0477·43-s − 0.739i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.1320302035\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1320302035\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 173. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 484T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.34e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 3.36e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 12.7iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 5.74e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 3.37e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 2.93e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 3.97e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 5.25e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 3.70e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 3.80e3T + 6.32e9T^{2} \) |
| 47 | \( 1 + 7.67e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 2.38e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 2.49e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 1.32e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + 1.68e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 5.31e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 2.36e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 3.51e4T + 2.43e11T^{2} \) |
| 83 | \( 1 + 1.09e4iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 1.29e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 3.21e5T + 8.32e11T^{2} \) |
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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
−10.27876645180557098915012920921, −9.877492568020404259030051185343, −8.985396480364529280009836643966, −7.37436806806884325193501357667, −7.05743335762494203404048075634, −6.27210631400804706917432337395, −5.03548391886707498778162653087, −3.72432729673021651103889937211, −2.87513367723826408985859619115, −2.02985928224993188831161217903,
0.04823688285102974160036775537, 0.49579283441015451900999534308, 2.05765000041784171102140571965, 3.26504092334992315888035816923, 4.33939507987719952692668634255, 5.33529193464473251429447855219, 6.17520176130796224768196222728, 7.22028686230430019203986614233, 8.321870104776695269627931387523, 9.094250275509000795959074762102