| L(s) = 1 | + (2.25 − 15.4i)3-s − 86.6i·5-s − 110. i·7-s + (−232. − 69.5i)9-s − 212.·11-s − 539.·13-s + (−1.33e3 − 195. i)15-s + 24.1i·17-s + 1.93e3i·19-s + (−1.70e3 − 249. i)21-s − 275.·23-s − 4.38e3·25-s + (−1.59e3 + 3.43e3i)27-s + 7.33e3i·29-s − 7.47e3i·31-s + ⋯ |
| L(s) = 1 | + (0.144 − 0.989i)3-s − 1.55i·5-s − 0.852i·7-s + (−0.958 − 0.286i)9-s − 0.528·11-s − 0.885·13-s + (−1.53 − 0.224i)15-s + 0.0202i·17-s + 1.22i·19-s + (−0.843 − 0.123i)21-s − 0.108·23-s − 1.40·25-s + (−0.421 + 0.906i)27-s + 1.62i·29-s − 1.39i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5545683038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5545683038\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.25 + 15.4i)T \) |
| good | 5 | \( 1 + 86.6iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 110. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 212.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 539.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 24.1iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.93e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 275.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.33e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 7.47e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 7.68e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.93e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 4.49e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 3.62e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.67e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.39e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.33e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.55e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 3.00e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.32e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.05e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 1.09e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.38e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 4.23e3T + 8.58e9T^{2} \) |
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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
−9.659999751232292391234255733941, −8.724920573902108094625521144181, −7.87143121841218598965289289099, −7.24536395608992363899796189905, −5.86113410896905994638477598581, −4.98649979096905674694393791873, −3.78574725264371397083321717423, −2.13902681069216277914163365540, −1.05413936570198072968761225091, −0.14507698845203705786446552158,
2.63267113056614842166981303956, 2.79417140413159322479452572300, 4.33392598031482556667865325691, 5.42285803007386920009283280848, 6.43426629126917853645416580140, 7.50450003559955032114664861742, 8.565631285254058738299885394798, 9.684957332248948210033862320554, 10.18639954369866259410790302006, 11.21706127781898360838404600992
