L(s) = 1 | + i·2-s + (1.41 + i)3-s − 4-s + (−2.12 + 0.707i)5-s + (−1 + 1.41i)6-s − i·8-s + (1.00 + 2.82i)9-s + (−0.707 − 2.12i)10-s + 2.82i·11-s + (−1.41 − i)12-s − 4.24·13-s + (−3.70 − 1.12i)15-s + 16-s + (−2.82 + 1.00i)18-s + (1 + 4.24i)19-s + (2.12 − 0.707i)20-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.816 + 0.577i)3-s − 0.5·4-s + (−0.948 + 0.316i)5-s + (−0.408 + 0.577i)6-s − 0.353i·8-s + (0.333 + 0.942i)9-s + (−0.223 − 0.670i)10-s + 0.852i·11-s + (−0.408 − 0.288i)12-s − 1.17·13-s + (−0.957 − 0.289i)15-s + 0.250·16-s + (−0.666 + 0.235i)18-s + (0.229 + 0.973i)19-s + (0.474 − 0.158i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0622i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0351910 + 1.13023i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0351910 + 1.13023i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-1.41 - i)T \) |
| 5 | \( 1 + (2.12 - 0.707i)T \) |
| 19 | \( 1 + (-1 - 4.24i)T \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 2.82iT - 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 + 4.24T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 4.24iT - 31T^{2} \) |
| 37 | \( 1 + 4.24T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 12.7iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−11.00658377494151154910804321557, −9.954624702111888940207214955766, −9.478973812713731920560581582203, −8.273043378636069290392746872663, −7.65220600239859273594896480910, −7.06533608656079239618441658940, −5.55709104935704091131346245771, −4.42887746046340257859912100042, −3.78894507740846766411173293716, −2.38062397866808396091169083585,
0.57445133394908193928306761427, 2.26944524551373368276288639410, 3.35421839489797830890578378539, 4.24980664160823953742159140562, 5.52106008817624816236439520742, 7.09320944726372468239357634226, 7.68889498634096737755418216616, 8.734138698715187983563810281700, 9.176070961563806686164754602401, 10.36700221175778596063267785054