| L(s) = 1 | − 168. i·5-s − 180.·7-s − 485. i·11-s + 3.11e3·13-s + 2.71e3i·17-s − 4.42e3·19-s − 2.32e4i·23-s − 1.29e4·25-s + 2.71e3i·29-s + 5.19e4·31-s + 3.05e4i·35-s + 8.12e4·37-s + 1.72e3i·41-s + 1.49e5·43-s − 1.32e5i·47-s + ⋯ |
| L(s) = 1 | − 1.35i·5-s − 0.527·7-s − 0.364i·11-s + 1.41·13-s + 0.553i·17-s − 0.644·19-s − 1.90i·23-s − 0.827·25-s + 0.111i·29-s + 1.74·31-s + 0.713i·35-s + 1.60·37-s + 0.0250i·41-s + 1.87·43-s − 1.27i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.004763634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.004763634\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 168. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 180.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 485. iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 3.11e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 2.71e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 4.42e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 2.32e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 2.71e3iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 5.19e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 8.12e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 1.72e3iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.49e5T + 6.32e9T^{2} \) |
| 47 | \( 1 + 1.32e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 2.16e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 9.87e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 1.97e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 2.99e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 3.88e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 2.33e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 6.22e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 3.89e4iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 3.88e4iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.00e6T + 8.32e11T^{2} \) |
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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
−9.248449426326532495637250984381, −8.566958180127736224244628546149, −8.062146108747577925225396298985, −6.43746648122883986841291826944, −5.97559009236825110042280300400, −4.65641827102308335780059634301, −3.98786392041609553118494878705, −2.62786203169559388486922928483, −1.19544757760235940545307121448, −0.48180019635429231915547943838,
1.09958993919252629780147871204, 2.51684223247330565995701995015, 3.31861526800040200989955098566, 4.28437316719177645285057565176, 5.83987133508109933685718627314, 6.44330637946034709346151269268, 7.30013047540052886997812990307, 8.191292801342598022501246627574, 9.424215568205659276406968742502, 10.02052337258810107632153756038
