| L(s) = 1 | + (−4.73 + 12.0i)2-s + (−20.1 − 1.51i)3-s + (−76.0 − 70.5i)4-s + (43.5 − 63.9i)5-s + (113. − 235. i)6-s + (342. + 10.5i)7-s + (464. − 223. i)8-s + (−316. − 47.7i)9-s + (564. + 827. i)10-s + (1.95e3 − 294. i)11-s + (1.42e3 + 1.53e3i)12-s + (−744. + 593. i)13-s + (−1.74e3 + 4.08e3i)14-s + (−974. + 1.22e3i)15-s + (2.25 + 30.0i)16-s + (304. − 988. i)17-s + ⋯ |
| L(s) = 1 | + (−0.591 + 1.50i)2-s + (−0.746 − 0.0559i)3-s + (−1.18 − 1.10i)4-s + (0.348 − 0.511i)5-s + (0.525 − 1.09i)6-s + (0.999 + 0.0306i)7-s + (0.906 − 0.436i)8-s + (−0.434 − 0.0655i)9-s + (0.564 + 0.827i)10-s + (1.46 − 0.221i)11-s + (0.825 + 0.889i)12-s + (−0.338 + 0.270i)13-s + (−0.637 + 1.48i)14-s + (−0.288 + 0.362i)15-s + (0.000550 + 0.00734i)16-s + (0.0620 − 0.201i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0669 - 0.997i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.0669 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.726211 + 0.776567i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.726211 + 0.776567i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-342. - 10.5i)T \) |
| good | 2 | \( 1 + (4.73 - 12.0i)T + (-46.9 - 43.5i)T^{2} \) |
| 3 | \( 1 + (20.1 + 1.51i)T + (720. + 108. i)T^{2} \) |
| 5 | \( 1 + (-43.5 + 63.9i)T + (-5.70e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-1.95e3 + 294. i)T + (1.69e6 - 5.22e5i)T^{2} \) |
| 13 | \( 1 + (744. - 593. i)T + (1.07e6 - 4.70e6i)T^{2} \) |
| 17 | \( 1 + (-304. + 988. i)T + (-1.99e7 - 1.35e7i)T^{2} \) |
| 19 | \( 1 + (2.14e3 - 1.23e3i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-2.38e3 + 735. i)T + (1.22e8 - 8.33e7i)T^{2} \) |
| 29 | \( 1 + (-7.85e3 - 3.43e4i)T + (-5.35e8 + 2.58e8i)T^{2} \) |
| 31 | \( 1 + (-2.13e4 - 1.23e4i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-4.57e4 + 4.24e4i)T + (1.91e8 - 2.55e9i)T^{2} \) |
| 41 | \( 1 + (2.14e4 + 4.45e4i)T + (-2.96e9 + 3.71e9i)T^{2} \) |
| 43 | \( 1 + (-1.07e5 - 5.19e4i)T + (3.94e9 + 4.94e9i)T^{2} \) |
| 47 | \( 1 + (4.02e3 + 1.58e3i)T + (7.90e9 + 7.33e9i)T^{2} \) |
| 53 | \( 1 + (2.53e4 + 2.35e4i)T + (1.65e9 + 2.21e10i)T^{2} \) |
| 59 | \( 1 + (1.59e5 + 2.33e5i)T + (-1.54e10 + 3.92e10i)T^{2} \) |
| 61 | \( 1 + (-1.82e5 - 1.96e5i)T + (-3.85e9 + 5.13e10i)T^{2} \) |
| 67 | \( 1 + (-3.15e4 + 5.47e4i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + (6.60e4 - 2.89e5i)T + (-1.15e11 - 5.55e10i)T^{2} \) |
| 73 | \( 1 + (-6.09e5 + 2.39e5i)T + (1.10e11 - 1.02e11i)T^{2} \) |
| 79 | \( 1 + (-5.87e4 - 1.01e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (5.64e5 + 4.50e5i)T + (7.27e10 + 3.18e11i)T^{2} \) |
| 89 | \( 1 + (3.26e4 - 2.16e5i)T + (-4.74e11 - 1.46e11i)T^{2} \) |
| 97 | \( 1 + 1.48e6iT - 8.32e11T^{2} \) |
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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
−14.62722371677939739804600169372, −14.18557843862843930459517044163, −12.23839971675080047154547172550, −11.11066866426845507831698311222, −9.286311293980442480241175714054, −8.501078928274646655807793777132, −7.00637530799014699360596170788, −5.88053728906150649218904905246, −4.82790730843941471497628775185, −1.01642170494629054106478936786,
0.915861736241945238962337190416, 2.45800332540438625990602483809, 4.36925696365405250658770898307, 6.27432871555656020436780268307, 8.304927270291283641800723931234, 9.639784780610533431138929439860, 10.73808691413774613906642256765, 11.53698027020675834549512613618, 12.20268775049854631781779515724, 13.85693461269565902386169928683
