| L(s) = 1 | + 1.19i·2-s + (−22.9 − 14.1i)3-s + 62.5·4-s − 242. i·5-s + (16.9 − 27.4i)6-s + 379.·7-s + 151. i·8-s + (327. + 651. i)9-s + 289.·10-s − 415. i·11-s + (−1.43e3 − 886. i)12-s − 2.68e3·13-s + 453. i·14-s + (−3.44e3 + 5.58e3i)15-s + 3.82e3·16-s − 6.15e3i·17-s + ⋯ |
| L(s) = 1 | + 0.149i·2-s + (−0.851 − 0.524i)3-s + 0.977·4-s − 1.94i·5-s + (0.0783 − 0.126i)6-s + 1.10·7-s + 0.295i·8-s + (0.448 + 0.893i)9-s + 0.289·10-s − 0.312i·11-s + (−0.832 − 0.513i)12-s − 1.22·13-s + 0.165i·14-s + (−1.02 + 1.65i)15-s + 0.933·16-s − 1.25i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.832400 - 1.49141i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.832400 - 1.49141i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (22.9 + 14.1i)T \) |
| 23 | \( 1 - 2.53e3iT \) |
| good | 2 | \( 1 - 1.19iT - 64T^{2} \) |
| 5 | \( 1 + 242. iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 379.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 415. iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 2.68e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 6.15e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 8.44e3T + 4.70e7T^{2} \) |
| 29 | \( 1 + 1.50e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 2.96e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 4.48e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 6.75e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 2.37e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 8.41e3iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 3.11e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 7.48e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 4.22e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 1.96e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 7.01e4iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 7.08e4T + 1.51e11T^{2} \) |
| 79 | \( 1 - 2.02e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 1.38e4iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 2.27e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 3.82e5T + 8.32e11T^{2} \) |
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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
−12.84207808571095951329589480884, −11.82036579333374998216931736934, −11.50196029262853085486936539902, −9.710196096920618254600694899600, −8.123430159496859120830866670065, −7.29892522351243186608073792677, −5.44114155253337806071374051639, −4.93541105454454904191644420710, −1.87173890837970847573821338110, −0.73594621411867571588042772897,
1.99954442917745362827365157167, 3.53862536511770281549257424454, 5.46395466565024012563607553362, 6.79504284556592163196035036999, 7.51941796684527321135921563096, 10.00427393911902155773854274933, 10.68462970877691024700571985422, 11.39957885464254867162955484709, 12.23366860427687877682746345796, 14.49129971572631780430271774938
