| L(s) = 1 | + (−5.28 + 13.4i)2-s + (−41.7 − 3.13i)3-s + (−106. − 98.5i)4-s + (0.579 − 0.849i)5-s + (262. − 545. i)6-s + (−337. − 63.0i)7-s + (1.05e3 − 507. i)8-s + (1.01e3 + 152. i)9-s + (8.37 + 12.2i)10-s + (−2.12e3 + 320. i)11-s + (4.12e3 + 4.45e3i)12-s + (45.0 − 35.9i)13-s + (2.62e3 − 4.20e3i)14-s + (−26.8 + 33.6i)15-s + (570. + 7.61e3i)16-s + (948. − 3.07e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.660 + 1.68i)2-s + (−1.54 − 0.115i)3-s + (−1.66 − 1.54i)4-s + (0.00463 − 0.00679i)5-s + (1.21 − 2.52i)6-s + (−0.982 − 0.183i)7-s + (2.05 − 0.991i)8-s + (1.39 + 0.209i)9-s + (0.00837 + 0.0122i)10-s + (−1.59 + 0.240i)11-s + (2.38 + 2.57i)12-s + (0.0205 − 0.0163i)13-s + (0.957 − 1.53i)14-s + (−0.00795 + 0.00998i)15-s + (0.139 + 1.85i)16-s + (0.193 − 0.625i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.171675 + 0.134550i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.171675 + 0.134550i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (337. + 63.0i)T \) |
| good | 2 | \( 1 + (5.28 - 13.4i)T + (-46.9 - 43.5i)T^{2} \) |
| 3 | \( 1 + (41.7 + 3.13i)T + (720. + 108. i)T^{2} \) |
| 5 | \( 1 + (-0.579 + 0.849i)T + (-5.70e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (2.12e3 - 320. i)T + (1.69e6 - 5.22e5i)T^{2} \) |
| 13 | \( 1 + (-45.0 + 35.9i)T + (1.07e6 - 4.70e6i)T^{2} \) |
| 17 | \( 1 + (-948. + 3.07e3i)T + (-1.99e7 - 1.35e7i)T^{2} \) |
| 19 | \( 1 + (1.01e4 - 5.83e3i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (6.96e3 - 2.14e3i)T + (1.22e8 - 8.33e7i)T^{2} \) |
| 29 | \( 1 + (3.70e3 + 1.62e4i)T + (-5.35e8 + 2.58e8i)T^{2} \) |
| 31 | \( 1 + (-4.97e4 - 2.87e4i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-2.39e4 + 2.21e4i)T + (1.91e8 - 2.55e9i)T^{2} \) |
| 41 | \( 1 + (3.63e3 + 7.53e3i)T + (-2.96e9 + 3.71e9i)T^{2} \) |
| 43 | \( 1 + (4.46e4 + 2.15e4i)T + (3.94e9 + 4.94e9i)T^{2} \) |
| 47 | \( 1 + (3.36e4 + 1.32e4i)T + (7.90e9 + 7.33e9i)T^{2} \) |
| 53 | \( 1 + (1.26e5 + 1.17e5i)T + (1.65e9 + 2.21e10i)T^{2} \) |
| 59 | \( 1 + (-1.39e5 - 2.04e5i)T + (-1.54e10 + 3.92e10i)T^{2} \) |
| 61 | \( 1 + (1.49e5 + 1.60e5i)T + (-3.85e9 + 5.13e10i)T^{2} \) |
| 67 | \( 1 + (-2.46e5 + 4.27e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + (-9.64e4 + 4.22e5i)T + (-1.15e11 - 5.55e10i)T^{2} \) |
| 73 | \( 1 + (3.63e5 - 1.42e5i)T + (1.10e11 - 1.02e11i)T^{2} \) |
| 79 | \( 1 + (-1.15e5 - 1.99e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-3.92e5 - 3.13e5i)T + (7.27e10 + 3.18e11i)T^{2} \) |
| 89 | \( 1 + (1.00e5 - 6.69e5i)T + (-4.74e11 - 1.46e11i)T^{2} \) |
| 97 | \( 1 - 4.87e5iT - 8.32e11T^{2} \) |
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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
−15.23969538590699972015160908891, −13.54213608299934624097103293503, −12.45801285674362502259009481233, −10.61605421089240585769752507839, −9.813988545206868418743419491863, −8.044098575940205047369418155813, −6.81714551105923215508355494749, −5.95247797282294040840062169472, −4.91344638857853441681609971993, −0.35205498636935314671116133135,
0.48788937740564675995300624519, 2.64594164032690242104055508737, 4.53830901607789340503268177222, 6.22771999124223353327394768093, 8.381393960245925677286000201258, 10.05745703598173061521397273439, 10.53071832423795298054751241140, 11.54259211143785859186731893214, 12.64743117120370600672301284628, 13.12972956343005899926633181351
