L(s) = 1 | + (−2.63 + 1.03i)2-s + (−5.55 − 5.55i)3-s + (5.87 − 5.43i)4-s + (−10.4 − 3.84i)5-s + (20.3 + 8.91i)6-s + (1.14 − 1.14i)7-s + (−9.86 + 20.3i)8-s + 34.8i·9-s + (31.6 − 0.706i)10-s − 27.0i·11-s + (−62.8 − 2.45i)12-s + (40.4 − 40.4i)13-s + (−1.83 + 4.18i)14-s + (37.0 + 79.7i)15-s + (5.00 − 63.8i)16-s + (−36.2 − 36.2i)17-s + ⋯ |
L(s) = 1 | + (−0.931 + 0.364i)2-s + (−1.06 − 1.06i)3-s + (0.734 − 0.678i)4-s + (−0.939 − 0.343i)5-s + (1.38 + 0.606i)6-s + (0.0616 − 0.0616i)7-s + (−0.436 + 0.899i)8-s + 1.28i·9-s + (0.999 − 0.0223i)10-s − 0.741i·11-s + (−1.51 − 0.0591i)12-s + (0.863 − 0.863i)13-s + (−0.0349 + 0.0798i)14-s + (0.637 + 1.37i)15-s + (0.0781 − 0.996i)16-s + (−0.517 − 0.517i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.582 + 0.812i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.164049 - 0.319364i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.164049 - 0.319364i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2.63 - 1.03i)T \) |
| 5 | \( 1 + (10.4 + 3.84i)T \) |
good | 3 | \( 1 + (5.55 + 5.55i)T + 27iT^{2} \) |
| 7 | \( 1 + (-1.14 + 1.14i)T - 343iT^{2} \) |
| 11 | \( 1 + 27.0iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (-40.4 + 40.4i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + (36.2 + 36.2i)T + 4.91e3iT^{2} \) |
| 19 | \( 1 + 56.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-54.9 - 54.9i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + 57.1iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 190. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (50.4 + 50.4i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 + 71.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + (-66.9 - 66.9i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + (-343. + 343. i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 + (-240. + 240. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 - 738.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 187.T + 2.26e5T^{2} \) |
| 67 | \( 1 + (576. - 576. i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 + 157. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-180. + 180. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 - 55.6T + 4.93e5T^{2} \) |
| 83 | \( 1 + (858. + 858. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 - 158. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (1.11e3 + 1.11e3i)T + 9.12e5iT^{2} \) |
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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
−17.48997318319960071637473934612, −16.47800671510062611458052932766, −15.36462035561976728671344274640, −13.19744468246127585404830681673, −11.70598815623819220203864742067, −10.90949228533530803316209282688, −8.525622281329031562241862057551, −7.26700314553084322443395864821, −5.80065954854311085639031138187, −0.61822113697246821664982706940,
4.15265650491696262188221241899, 6.74045873526998573045375295989, 8.770884583890854202863449242969, 10.43753228466772904877410609527, 11.18796831169505155822348266419, 12.30207317086056224538366832587, 15.15231506827804522051301872298, 16.02995745916686105068772628437, 16.98303678304147708678869158970, 18.17687645615785436054919506012