| L(s) = 1 | + (7.38 + 3.07i)2-s + (45.0 + 45.4i)4-s + 70.4i·5-s + 87.7i·7-s + (193. + 474. i)8-s + (−216. + 520. i)10-s − 407.·11-s + 1.89e3i·13-s + (−270. + 648. i)14-s + (−33.3 + 4.09e3i)16-s − 2.51e3·17-s + 437.·19-s + (−3.20e3 + 3.17e3i)20-s + (−3.00e3 − 1.25e3i)22-s + 1.40e4i·23-s + ⋯ |
| L(s) = 1 | + (0.923 + 0.384i)2-s + (0.704 + 0.709i)4-s + 0.563i·5-s + 0.255i·7-s + (0.377 + 0.926i)8-s + (−0.216 + 0.520i)10-s − 0.305·11-s + 0.861i·13-s + (−0.0984 + 0.236i)14-s + (−0.00813 + 0.999i)16-s − 0.511·17-s + 0.0638·19-s + (−0.400 + 0.396i)20-s + (−0.282 − 0.117i)22-s + 1.15i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.377 - 0.926i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.377 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.64964 + 2.45265i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64964 + 2.45265i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-7.38 - 3.07i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 70.4iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 87.7iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 407.T + 1.77e6T^{2} \) |
| 13 | \( 1 - 1.89e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 2.51e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 437.T + 4.70e7T^{2} \) |
| 23 | \( 1 - 1.40e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 3.15e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 3.84e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 5.50e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 4.57e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 8.99e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 3.02e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 4.40e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 2.74e5T + 4.21e10T^{2} \) |
| 61 | \( 1 + 2.33e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 2.75e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 4.81e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 2.80e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 7.01e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 7.40e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 1.17e6T + 4.96e11T^{2} \) |
| 97 | \( 1 + 1.05e5T + 8.32e11T^{2} \) |
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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
−13.85780640312740677379721639802, −12.83239254315366376001633475807, −11.66805911956825640494506824061, −10.77254045966502311908719656752, −9.063241238320658049412752884594, −7.55156215210675165893484464326, −6.53838938106840925267224835946, −5.22280384240142119944156020881, −3.72699947695781505692636205627, −2.24537272006411974609572867987,
0.855133074330394161431088694450, 2.68124920119959118017348030425, 4.30087791122255318541025913355, 5.44130283573802762530882280515, 6.84460487494814735231035438375, 8.408668938014139653527798369705, 10.01631348964423738193170769003, 10.95059106404043526412340244741, 12.24578914763009237171598692023, 13.00606057924039575361838175485
