| L(s) = 1 | + (25.5 − 8.72i)3-s + 169. i·5-s − 431.·7-s + (576. − 445. i)9-s − 1.07e3i·11-s + 639.·13-s + (1.47e3 + 4.31e3i)15-s − 2.38e3i·17-s − 7.14e3·19-s + (−1.10e4 + 3.76e3i)21-s − 2.53e3i·23-s − 1.29e4·25-s + (1.08e4 − 1.64e4i)27-s − 1.92e4i·29-s − 1.37e3·31-s + ⋯ |
| L(s) = 1 | + (0.946 − 0.323i)3-s + 1.35i·5-s − 1.25·7-s + (0.791 − 0.611i)9-s − 0.805i·11-s + 0.291·13-s + (0.437 + 1.27i)15-s − 0.485i·17-s − 1.04·19-s + (−1.19 + 0.406i)21-s − 0.208i·23-s − 0.829·25-s + (0.550 − 0.834i)27-s − 0.790i·29-s − 0.0461·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.323 + 0.946i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.323 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.984175185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.984175185\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-25.5 + 8.72i)T \) |
| 23 | \( 1 + 2.53e3iT \) |
| good | 5 | \( 1 - 169. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 431.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 1.07e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 639.T + 4.82e6T^{2} \) |
| 17 | \( 1 + 2.38e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 7.14e3T + 4.70e7T^{2} \) |
| 29 | \( 1 + 1.92e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 1.37e3T + 8.87e8T^{2} \) |
| 37 | \( 1 - 6.19e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 2.33e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.23e5T + 6.32e9T^{2} \) |
| 47 | \( 1 + 1.44e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 7.71e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 1.62e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 2.83e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 4.25e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 1.68e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 1.61e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 1.16e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 3.99e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 4.98e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 5.64e5T + 8.32e11T^{2} \) |
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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
−10.50117221464551518880161476841, −9.713213065938767739893225934927, −8.755654586333359688165096912281, −7.65693043048367885426223632828, −6.66514199174920132107451911773, −6.14004157180416323349305339780, −3.95868234700895203091472384023, −3.09385122977575520812535020035, −2.36018989876017985174056864899, −0.45027450388024630753620973210,
1.17174888231249787908581040322, 2.50202882305380532650057982505, 3.86069924019758651905278361406, 4.63424804910629218187015574433, 6.01549628021783324387932836696, 7.29305571895070562905964904992, 8.401230285682518883093967144896, 9.154872690291521930208153724058, 9.745030178369679691045561773324, 10.77400385409999621147953697763
