| L(s) = 1 | + (−7.11 + 3.65i)2-s + (37.2 − 52.0i)4-s + 87.0i·5-s − 355. i·7-s + (−74.2 + 506. i)8-s + (−318. − 619. i)10-s + 185.·11-s + 2.32e3i·13-s + (1.30e3 + 2.52e3i)14-s + (−1.32e3 − 3.87e3i)16-s − 7.53e3·17-s − 807.·19-s + (4.53e3 + 3.24e3i)20-s + (−1.31e3 + 677. i)22-s − 1.51e4i·23-s + ⋯ |
| L(s) = 1 | + (−0.889 + 0.457i)2-s + (0.581 − 0.813i)4-s + 0.696i·5-s − 1.03i·7-s + (−0.145 + 0.989i)8-s + (−0.318 − 0.619i)10-s + 0.139·11-s + 1.05i·13-s + (0.474 + 0.921i)14-s + (−0.323 − 0.946i)16-s − 1.53·17-s − 0.117·19-s + (0.566 + 0.405i)20-s + (−0.123 + 0.0636i)22-s − 1.24i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.145 + 0.989i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.145 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.342406 - 0.396290i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.342406 - 0.396290i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (7.11 - 3.65i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 87.0iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 355. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 185.T + 1.77e6T^{2} \) |
| 13 | \( 1 - 2.32e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 7.53e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 807.T + 4.70e7T^{2} \) |
| 23 | \( 1 + 1.51e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 3.24e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 4.41e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 5.90e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 8.71e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 1.24e5T + 6.32e9T^{2} \) |
| 47 | \( 1 - 6.49e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.54e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 4.15e4T + 4.21e10T^{2} \) |
| 61 | \( 1 - 1.08e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 4.82e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 4.18e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 5.46e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 5.38e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 4.43e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 1.51e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + 1.18e6T + 8.32e11T^{2} \) |
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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
−13.42682373377700169012349516862, −11.50734916910349609106474573749, −10.74794969204687220120102532887, −9.701921306512951323638768210634, −8.448822951623008944274954483138, −7.06723468499504435293591129240, −6.44068538620986660854416789360, −4.34930686154595047914194422073, −2.16991901860858429496125865488, −0.26763700932196531077639297515,
1.51703529754358571139300671121, 3.10604065573940783002504525816, 5.13404824672784591133533031596, 6.81171182412662356553424556334, 8.440990607259896795662533099473, 8.943804912569054879731346983615, 10.27154240480498411342511642857, 11.48300171113560443313532824971, 12.44348711572527621215563434409, 13.27720060155776106031311952959
