| L(s) = 1 | + (1.55 − 3.95i)2-s + (1.47 + 0.110i)3-s + (33.6 + 31.2i)4-s + (−51.2 + 75.1i)5-s + (2.73 − 5.66i)6-s + (−331. + 89.3i)7-s + (420. − 202. i)8-s + (−718. − 108. i)9-s + (217. + 319. i)10-s + (−2.47e3 + 373. i)11-s + (46.3 + 49.9i)12-s + (1.81e3 − 1.44e3i)13-s + (−160. + 1.44e3i)14-s + (−83.9 + 105. i)15-s + (71.5 + 954. i)16-s + (−1.70e3 + 5.51e3i)17-s + ⋯ |
| L(s) = 1 | + (0.193 − 0.494i)2-s + (0.0547 + 0.00409i)3-s + (0.526 + 0.488i)4-s + (−0.409 + 0.600i)5-s + (0.0126 − 0.0262i)6-s + (−0.965 + 0.260i)7-s + (0.821 − 0.395i)8-s + (−0.985 − 0.148i)9-s + (0.217 + 0.319i)10-s + (−1.86 + 0.280i)11-s + (0.0267 + 0.0288i)12-s + (0.824 − 0.657i)13-s + (−0.0584 + 0.527i)14-s + (−0.0248 + 0.0311i)15-s + (0.0174 + 0.233i)16-s + (−0.346 + 1.12i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.609 - 0.792i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.609 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.350891 + 0.712400i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.350891 + 0.712400i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (331. - 89.3i)T \) |
| good | 2 | \( 1 + (-1.55 + 3.95i)T + (-46.9 - 43.5i)T^{2} \) |
| 3 | \( 1 + (-1.47 - 0.110i)T + (720. + 108. i)T^{2} \) |
| 5 | \( 1 + (51.2 - 75.1i)T + (-5.70e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (2.47e3 - 373. i)T + (1.69e6 - 5.22e5i)T^{2} \) |
| 13 | \( 1 + (-1.81e3 + 1.44e3i)T + (1.07e6 - 4.70e6i)T^{2} \) |
| 17 | \( 1 + (1.70e3 - 5.51e3i)T + (-1.99e7 - 1.35e7i)T^{2} \) |
| 19 | \( 1 + (3.47e3 - 2.00e3i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-1.03e4 + 3.19e3i)T + (1.22e8 - 8.33e7i)T^{2} \) |
| 29 | \( 1 + (-6.95e3 - 3.04e4i)T + (-5.35e8 + 2.58e8i)T^{2} \) |
| 31 | \( 1 + (4.10e4 + 2.37e4i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-2.75e4 + 2.55e4i)T + (1.91e8 - 2.55e9i)T^{2} \) |
| 41 | \( 1 + (1.60e4 + 3.33e4i)T + (-2.96e9 + 3.71e9i)T^{2} \) |
| 43 | \( 1 + (-3.48e4 - 1.67e4i)T + (3.94e9 + 4.94e9i)T^{2} \) |
| 47 | \( 1 + (-1.88e4 - 7.38e3i)T + (7.90e9 + 7.33e9i)T^{2} \) |
| 53 | \( 1 + (-1.30e4 - 1.21e4i)T + (1.65e9 + 2.21e10i)T^{2} \) |
| 59 | \( 1 + (-1.12e5 - 1.65e5i)T + (-1.54e10 + 3.92e10i)T^{2} \) |
| 61 | \( 1 + (2.74e5 + 2.96e5i)T + (-3.85e9 + 5.13e10i)T^{2} \) |
| 67 | \( 1 + (2.33e5 - 4.05e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + (4.92e4 - 2.15e5i)T + (-1.15e11 - 5.55e10i)T^{2} \) |
| 73 | \( 1 + (1.72e5 - 6.75e4i)T + (1.10e11 - 1.02e11i)T^{2} \) |
| 79 | \( 1 + (-1.31e5 - 2.26e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (5.52e5 + 4.40e5i)T + (7.27e10 + 3.18e11i)T^{2} \) |
| 89 | \( 1 + (1.97e5 - 1.31e6i)T + (-4.74e11 - 1.46e11i)T^{2} \) |
| 97 | \( 1 + 5.16e5iT - 8.32e11T^{2} \) |
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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
−14.95913885706770120682869500605, −13.08682128271557906383445686851, −12.67820773532479958022490613646, −11.02556698425974184342702369781, −10.55883104512584515417138857412, −8.555091990221773025825347179987, −7.32245858272071747776276900205, −5.83929681218794331914487508615, −3.49555039882035453771131959653, −2.60290538687678546306565400943,
0.30071697882131132949025198384, 2.74587083714701679670374206082, 4.93604130694007757025913126049, 6.17223756618565561752340724020, 7.56514945788017305009050980559, 8.892577414069352856883712991712, 10.51324599385147709839453408462, 11.47263231388551005340769902356, 13.10349594317948427491226241512, 13.87163619209111942083537533301
