| L(s) = 1 | + (−11.0 − 11.0i)3-s + (−45.7 − 45.7i)5-s + 565.·7-s + 242. i·9-s + (1.29e3 − 1.29e3i)11-s + (2.88e3 − 2.88e3i)13-s + 1.00e3i·15-s − 662.·17-s + (1.94e3 + 1.94e3i)19-s + (−6.23e3 − 6.23e3i)21-s + 1.58e4·23-s − 1.14e4i·25-s + (2.67e3 − 2.67e3i)27-s + (−1.15e4 + 1.15e4i)29-s + 3.90e4i·31-s + ⋯ |
| L(s) = 1 | + (−0.408 − 0.408i)3-s + (−0.366 − 0.366i)5-s + 1.64·7-s + 0.333i·9-s + (0.970 − 0.970i)11-s + (1.31 − 1.31i)13-s + 0.299i·15-s − 0.134·17-s + (0.282 + 0.282i)19-s + (−0.673 − 0.673i)21-s + 1.30·23-s − 0.731i·25-s + (0.136 − 0.136i)27-s + (−0.473 + 0.473i)29-s + 1.31i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.480 + 0.876i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.480 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.804248676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.804248676\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (11.0 + 11.0i)T \) |
| good | 5 | \( 1 + (45.7 + 45.7i)T + 1.56e4iT^{2} \) |
| 7 | \( 1 - 565.T + 1.17e5T^{2} \) |
| 11 | \( 1 + (-1.29e3 + 1.29e3i)T - 1.77e6iT^{2} \) |
| 13 | \( 1 + (-2.88e3 + 2.88e3i)T - 4.82e6iT^{2} \) |
| 17 | \( 1 + 662.T + 2.41e7T^{2} \) |
| 19 | \( 1 + (-1.94e3 - 1.94e3i)T + 4.70e7iT^{2} \) |
| 23 | \( 1 - 1.58e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + (1.15e4 - 1.15e4i)T - 5.94e8iT^{2} \) |
| 31 | \( 1 - 3.90e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + (-5.13e4 - 5.13e4i)T + 2.56e9iT^{2} \) |
| 41 | \( 1 - 6.65e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + (-6.89e4 + 6.89e4i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 - 1.07e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + (-1.08e5 - 1.08e5i)T + 2.21e10iT^{2} \) |
| 59 | \( 1 + (6.78e4 - 6.78e4i)T - 4.21e10iT^{2} \) |
| 61 | \( 1 + (-9.92e4 + 9.92e4i)T - 5.15e10iT^{2} \) |
| 67 | \( 1 + (1.05e5 + 1.05e5i)T + 9.04e10iT^{2} \) |
| 71 | \( 1 + 2.66e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 2.67e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 5.57e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (2.36e5 + 2.36e5i)T + 3.26e11iT^{2} \) |
| 89 | \( 1 - 4.19e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 3.93e5T + 8.32e11T^{2} \) |
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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
−10.60738829914905439490107980072, −8.847174292497571475186830770162, −8.365686536898221711827489511367, −7.54452774427168323590352414622, −6.24097861913854490994772994973, −5.37503076107044573715971815039, −4.39078208509864819532518839880, −3.12509869170545662738426796545, −1.26573849975025041230519797288, −0.958094853778073314927054761381,
1.06087537984157374618709754645, 2.03337208767693768433368786571, 3.92656276485102156264177695065, 4.42043459050336429066470793914, 5.58917023542823843020409774637, 6.80794948508180177655174275998, 7.57873470444875748320967077963, 8.821652796633237250042945769726, 9.430377625107190657469612134242, 10.86000050986724112299916389034
