| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.320072 - 1.00703i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.320072 - 1.00703i\) |
| \(L(4)\) |
|
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
−12.62076208536852622578125508595, −12.27431051024340862529099428678, −10.71518727278514753119837884900, −9.374600987283723847700072270343, −8.445691679005783786012351514839, −7.15307245468968270485602888964, −5.27094362804046630203484511880, −4.43556412831646649653110924514, −2.10550101352091554343614675603, −0.38573563338861791928561612401,
2.10397485713043678910479358031, 3.59016640170314890797546587957, 5.41001402871559987587909999003, 6.88880699347434111842344994812, 7.82326395834155808051044780065, 9.469065101840569423293882422599, 10.65738050606621781360356316639, 11.42563117445362362948944248471, 12.74229441827765372571429409911, 14.20461097929388805340012264071
