L(s) = 1 | + (2.44 − 1.41i)2-s + (−5.16 + 0.563i)3-s + (3.98 − 6.93i)4-s + (13.1 − 13.1i)5-s + (−11.8 + 8.70i)6-s − 13.2·7-s + (−0.0933 − 22.6i)8-s + (26.3 − 5.82i)9-s + (13.5 − 50.8i)10-s + (24.0 + 24.0i)11-s + (−16.6 + 38.0i)12-s + (−30.7 + 30.7i)13-s + (−32.4 + 18.8i)14-s + (−60.5 + 75.4i)15-s + (−32.3 − 55.2i)16-s + 56.8i·17-s + ⋯ |
L(s) = 1 | + (0.865 − 0.501i)2-s + (−0.994 + 0.108i)3-s + (0.497 − 0.867i)4-s + (1.17 − 1.17i)5-s + (−0.805 + 0.592i)6-s − 0.716·7-s + (−0.00412 − 0.999i)8-s + (0.976 − 0.215i)9-s + (0.428 − 1.60i)10-s + (0.660 + 0.660i)11-s + (−0.400 + 0.916i)12-s + (−0.656 + 0.656i)13-s + (−0.620 + 0.359i)14-s + (−1.04 + 1.29i)15-s + (−0.504 − 0.863i)16-s + 0.811i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.41430 - 1.10414i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.41430 - 1.10414i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.44 + 1.41i)T \) |
| 3 | \( 1 + (5.16 - 0.563i)T \) |
good | 5 | \( 1 + (-13.1 + 13.1i)T - 125iT^{2} \) |
| 7 | \( 1 + 13.2T + 343T^{2} \) |
| 11 | \( 1 + (-24.0 - 24.0i)T + 1.33e3iT^{2} \) |
| 13 | \( 1 + (30.7 - 30.7i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 - 56.8iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (-74.2 - 74.2i)T + 6.85e3iT^{2} \) |
| 23 | \( 1 + 21.7iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-102. - 102. i)T + 2.43e4iT^{2} \) |
| 31 | \( 1 + 219. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (-83.3 - 83.3i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 + 7.92T + 6.89e4T^{2} \) |
| 43 | \( 1 + (153. - 153. i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + 208.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-390. + 390. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (-221. - 221. i)T + 2.05e5iT^{2} \) |
| 61 | \( 1 + (416. - 416. i)T - 2.26e5iT^{2} \) |
| 67 | \( 1 + (284. + 284. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 - 26.9iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 839. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 556. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-400. + 400. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 + 974.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.55e3T + 9.12e5T^{2} \) |
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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
−14.74805161059845795982663842134, −13.39185202908982690029209883197, −12.57185577903592939574002402176, −11.82986154731270806599662107726, −10.05936999562042960196534452663, −9.541323574339827701937756214519, −6.61161724814961357689977693713, −5.59078327073439414780293152790, −4.38598742539127202804887703206, −1.52110453444286401239134140330,
2.97404973827269097712212555463, 5.29116845523752135358516845711, 6.37037263868638358798760615594, 7.13606988842825956953039347709, 9.671608461219558005335788097386, 10.90335148650892294676038849463, 12.02945322916397318729170660761, 13.35276929340758508427934810146, 14.07220592547578974816856524449, 15.40127832885623803322530300329