| L(s) = 1 | − 15.5·3-s − 20i·5-s − 529. i·7-s + 243·9-s − 435.·11-s + 341. i·13-s + 311. i·15-s + 7.68e3·17-s + 4.30e3·19-s + 8.25e3i·21-s + 3.17e3i·23-s + 1.52e4·25-s − 3.78e3·27-s − 1.94e4i·29-s − 1.52e4i·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.160i·5-s − 1.54i·7-s + 0.333·9-s − 0.327·11-s + 0.155i·13-s + 0.0923i·15-s + 1.56·17-s + 0.626·19-s + 0.891i·21-s + 0.260i·23-s + 0.974·25-s − 0.192·27-s − 0.795i·29-s − 0.513i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.730562080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.730562080\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 15.5T \) |
| good | 5 | \( 1 + 20iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 529. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 435.T + 1.77e6T^{2} \) |
| 13 | \( 1 - 341. iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 7.68e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 4.30e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 3.17e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 1.94e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 1.52e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 6.19e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 3.37e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 9.93e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.77e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 2.24e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 1.99e5T + 4.21e10T^{2} \) |
| 61 | \( 1 + 4.56e4iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 4.96e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 4.52e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 3.94e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 5.71e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 3.24e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 7.58e5T + 4.96e11T^{2} \) |
| 97 | \( 1 - 2.50e4T + 8.32e11T^{2} \) |
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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
−10.22041612876917935968908155327, −9.461301289789904600243672157939, −7.924919174889369336349607292920, −7.38670496518978375832258241365, −6.30674595743176566940309489442, −5.19689678640519734487301646742, −4.24494802302882985728911973214, −3.16389418940789646890479026021, −1.31007729172922154673166727987, −0.55058827640001653518760982867,
0.986739471808280069230934711453, 2.39211733015478362630651512981, 3.44446077065807754147109009133, 5.17941494125757559074590864494, 5.52979193857014294458050617909, 6.67104866403353069358787081161, 7.78235953163170658651963127595, 8.777021029806436357859163039049, 9.671534122962422588887126681022, 10.61639841999517415556252165773
