| L(s) = 1 | + 15.5·3-s − 20i·5-s − 155. i·7-s + 243·9-s − 2.30e3·11-s − 3.22e3i·13-s − 311. i·15-s − 6.56e3·17-s + 3.92e3·19-s − 2.42e3i·21-s + 1.73e4i·23-s + 1.52e4·25-s + 3.78e3·27-s + 4.47e4i·29-s − 3.06e4i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.160i·5-s − 0.454i·7-s + 0.333·9-s − 1.73·11-s − 1.46i·13-s − 0.0923i·15-s − 1.33·17-s + 0.572·19-s − 0.262i·21-s + 1.42i·23-s + 0.974·25-s + 0.192·27-s + 1.83i·29-s − 1.02i·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
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.172084107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.172084107\) |
| \(L(4)\) |
|
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 | \( 1 + 155. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 2.30e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + 3.22e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 6.56e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 3.92e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 1.73e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 4.47e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 3.06e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 4.41e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 9.00e3T + 4.75e9T^{2} \) |
| 43 | \( 1 + 2.52e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.75e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 9.62e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 3.50e4T + 4.21e10T^{2} \) |
| 61 | \( 1 - 8.61e4iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 4.24e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 3.65e4iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 3.75e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 5.20e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 5.94e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 9.01e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + 1.25e6T + 8.32e11T^{2} \) |
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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
−10.54312038813479200329867768311, −9.701721159413026310069349610363, −8.591316573801081200065568195854, −7.81220367811376445095626079016, −7.09270205406052130243783669944, −5.57494729999077230555562229734, −4.79511067321974240198204061123, −3.35803496105749866866115515188, −2.55709092696575739361760104457, −1.04988784972774636629409630315,
0.25292739133215055809237351601, 2.12003606621360491855343777696, 2.68315984098208017332870987040, 4.19330390412043464449932442340, 5.10863541598315236194614165245, 6.44441896204282826724971970436, 7.30601346985704866631905527000, 8.428933682126227842385380298405, 9.003462346386623654841449803942, 10.11777183586637945108902788739
