| L(s) = 1 | + (13.0 + 2.30i)3-s + (175. + 147. i)5-s + (274. + 475. i)7-s + (−519. − 189. i)9-s + (322. − 559. i)11-s + (−3.84e3 + 678. i)13-s + (1.95e3 + 2.32e3i)15-s + (7.16e3 − 2.60e3i)17-s + (−5.40e3 + 4.21e3i)19-s + (2.49e3 + 6.85e3i)21-s + (−95.2 + 79.9i)23-s + (6.37e3 + 3.61e4i)25-s + (−1.47e4 − 8.51e3i)27-s + (2.35e3 − 6.48e3i)29-s + (1.93e4 − 1.11e4i)31-s + ⋯ |
| L(s) = 1 | + (0.484 + 0.0854i)3-s + (1.40 + 1.17i)5-s + (0.801 + 1.38i)7-s + (−0.712 − 0.259i)9-s + (0.242 − 0.420i)11-s + (−1.75 + 0.308i)13-s + (0.578 + 0.689i)15-s + (1.45 − 0.530i)17-s + (−0.788 + 0.615i)19-s + (0.269 + 0.740i)21-s + (−0.00783 + 0.00657i)23-s + (0.407 + 2.31i)25-s + (−0.748 − 0.432i)27-s + (0.0967 − 0.265i)29-s + (0.648 − 0.374i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0896 - 0.995i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.0896 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.95431 + 1.78630i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.95431 + 1.78630i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (5.40e3 - 4.21e3i)T \) |
| good | 3 | \( 1 + (-13.0 - 2.30i)T + (685. + 249. i)T^{2} \) |
| 5 | \( 1 + (-175. - 147. i)T + (2.71e3 + 1.53e4i)T^{2} \) |
| 7 | \( 1 + (-274. - 475. i)T + (-5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-322. + 559. i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + (3.84e3 - 678. i)T + (4.53e6 - 1.65e6i)T^{2} \) |
| 17 | \( 1 + (-7.16e3 + 2.60e3i)T + (1.84e7 - 1.55e7i)T^{2} \) |
| 23 | \( 1 + (95.2 - 79.9i)T + (2.57e7 - 1.45e8i)T^{2} \) |
| 29 | \( 1 + (-2.35e3 + 6.48e3i)T + (-4.55e8 - 3.82e8i)T^{2} \) |
| 31 | \( 1 + (-1.93e4 + 1.11e4i)T + (4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 - 1.15e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (-8.40e4 - 1.48e4i)T + (4.46e9 + 1.62e9i)T^{2} \) |
| 43 | \( 1 + (-2.28e4 - 1.91e4i)T + (1.09e9 + 6.22e9i)T^{2} \) |
| 47 | \( 1 + (-3.03e4 - 1.10e4i)T + (8.25e9 + 6.92e9i)T^{2} \) |
| 53 | \( 1 + (-5.08e4 - 6.06e4i)T + (-3.84e9 + 2.18e10i)T^{2} \) |
| 59 | \( 1 + (9.73e4 + 2.67e5i)T + (-3.23e10 + 2.71e10i)T^{2} \) |
| 61 | \( 1 + (-2.19e5 + 1.84e5i)T + (8.94e9 - 5.07e10i)T^{2} \) |
| 67 | \( 1 + (5.18e4 - 1.42e5i)T + (-6.92e10 - 5.81e10i)T^{2} \) |
| 71 | \( 1 + (-4.38e5 + 5.22e5i)T + (-2.22e10 - 1.26e11i)T^{2} \) |
| 73 | \( 1 + (2.41e4 - 1.36e5i)T + (-1.42e11 - 5.17e10i)T^{2} \) |
| 79 | \( 1 + (5.91e5 + 1.04e5i)T + (2.28e11 + 8.31e10i)T^{2} \) |
| 83 | \( 1 + (-7.71e4 - 1.33e5i)T + (-1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 + (-2.91e5 + 5.13e4i)T + (4.67e11 - 1.69e11i)T^{2} \) |
| 97 | \( 1 + (-4.00e5 - 1.09e6i)T + (-6.38e11 + 5.35e11i)T^{2} \) |
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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
−14.14591667075713138842397075415, −12.35719031779521859095148006757, −11.38588479887525309173035503805, −9.956644950953571786062639886288, −9.225324836943254149243388693578, −7.85663688486390296259933227866, −6.21943906491575329076768478095, −5.34677422523302531450862802942, −2.85618492494420224968691301236, −2.15099583277229307746442429384,
0.965686617886498329749515798636, 2.28770040135263499216421699991, 4.52809502650958083988480778546, 5.52498817195594109798921432729, 7.36625034011944676448079181386, 8.465324784776576273527367119020, 9.680167250786553100260128093489, 10.49754662318904523631587833290, 12.22240593571731430160288005527, 13.17686649703916821962989727894
