L(s) = 1 | + (−1.73 − i)2-s + (4.53 + 7.86i)3-s + (1.99 + 3.46i)4-s + (4.92 − 2.84i)5-s − 18.1i·6-s + (4.51 + 7.81i)7-s − 7.99i·8-s + (−27.7 + 48.0i)9-s − 11.3·10-s − 13.9·11-s + (−18.1 + 31.4i)12-s + (−15.4 + 8.90i)13-s − 18.0i·14-s + (44.7 + 25.8i)15-s + (−8 + 13.8i)16-s + (104. + 60.5i)17-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.873 + 1.51i)3-s + (0.249 + 0.433i)4-s + (0.440 − 0.254i)5-s − 1.23i·6-s + (0.243 + 0.422i)7-s − 0.353i·8-s + (−1.02 + 1.77i)9-s − 0.359·10-s − 0.382·11-s + (−0.436 + 0.756i)12-s + (−0.329 + 0.189i)13-s − 0.344i·14-s + (0.770 + 0.444i)15-s + (−0.125 + 0.216i)16-s + (1.49 + 0.863i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.231 - 0.972i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.231 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.17562 + 0.928982i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.17562 + 0.928982i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.73 + i)T \) |
| 37 | \( 1 + (-124. - 187. i)T \) |
good | 3 | \( 1 + (-4.53 - 7.86i)T + (-13.5 + 23.3i)T^{2} \) |
| 5 | \( 1 + (-4.92 + 2.84i)T + (62.5 - 108. i)T^{2} \) |
| 7 | \( 1 + (-4.51 - 7.81i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + 13.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + (15.4 - 8.90i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-104. - 60.5i)T + (2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (9.24 - 5.33i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + 139. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 103. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 221. iT - 2.97e4T^{2} \) |
| 41 | \( 1 + (-60.1 - 104. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + 434. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 306.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-334. + 578. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-719. - 415. i)T + (1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (315. - 181. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (305. + 529. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (251. + 435. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + 507.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (809. - 467. i)T + (2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (478. - 828. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-584. - 337. i)T + (3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 532. iT - 9.12e5T^{2} \) |
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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
−14.64879498466234091968048986946, −13.32683748137160697927058414696, −11.86391135659979343666280307592, −10.43613533801045145263615184482, −9.862524404174304220061138223272, −8.831052172526018092123953044545, −7.947744406888953885438101767816, −5.51489662832577369850413521249, −3.98695574288415705467409393833, −2.44179116901394898526640146979,
1.20962491063061255987999815304, 2.80800372424722926486713398271, 5.76600195161456143306620416817, 7.24632615190140165235261154719, 7.70321261917611688505322007865, 9.005292294738825273630110556190, 10.21906948251716107749466268732, 11.81367577496158850234548808089, 12.93854346776236457011769108291, 14.08119615974098226559178508921