L(s) = 1 | + (−0.908 + 3.39i)2-s + (2.68 − 4.44i)3-s + (−3.73 − 2.15i)4-s + (12.1 + 12.1i)5-s + (12.6 + 13.1i)6-s + (−3.81 + 1.02i)7-s + (−9.13 + 9.13i)8-s + (−12.5 − 23.8i)9-s + (−52.1 + 30.1i)10-s + (7.61 + 2.03i)11-s + (−19.6 + 10.8i)12-s + (44.8 − 13.7i)13-s − 13.8i·14-s + (86.6 − 21.3i)15-s + (−39.9 − 69.1i)16-s + (−17.2 + 29.8i)17-s + ⋯ |
L(s) = 1 | + (−0.321 + 1.19i)2-s + (0.516 − 0.856i)3-s + (−0.467 − 0.269i)4-s + (1.08 + 1.08i)5-s + (0.859 + 0.894i)6-s + (−0.206 + 0.0552i)7-s + (−0.403 + 0.403i)8-s + (−0.465 − 0.885i)9-s + (−1.65 + 0.952i)10-s + (0.208 + 0.0559i)11-s + (−0.472 + 0.260i)12-s + (0.955 − 0.293i)13-s − 0.264i·14-s + (1.49 − 0.368i)15-s + (−0.624 − 1.08i)16-s + (−0.246 + 0.426i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.267 - 0.963i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.267 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.14011 + 0.866356i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.14011 + 0.866356i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.68 + 4.44i)T \) |
| 13 | \( 1 + (-44.8 + 13.7i)T \) |
good | 2 | \( 1 + (0.908 - 3.39i)T + (-6.92 - 4i)T^{2} \) |
| 5 | \( 1 + (-12.1 - 12.1i)T + 125iT^{2} \) |
| 7 | \( 1 + (3.81 - 1.02i)T + (297. - 171.5i)T^{2} \) |
| 11 | \( 1 + (-7.61 - 2.03i)T + (1.15e3 + 665.5i)T^{2} \) |
| 17 | \( 1 + (17.2 - 29.8i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (26.2 + 97.9i)T + (-5.94e3 + 3.42e3i)T^{2} \) |
| 23 | \( 1 + (98.5 + 170. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-60.6 + 35.0i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (70.3 - 70.3i)T - 2.97e4iT^{2} \) |
| 37 | \( 1 + (-38.6 + 144. i)T + (-4.38e4 - 2.53e4i)T^{2} \) |
| 41 | \( 1 + (82.0 - 306. i)T + (-5.96e4 - 3.44e4i)T^{2} \) |
| 43 | \( 1 + (-410. - 237. i)T + (3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (150. - 150. i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 - 401. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (146. + 545. i)T + (-1.77e5 + 1.02e5i)T^{2} \) |
| 61 | \( 1 + (-272. + 471. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-490. - 131. i)T + (2.60e5 + 1.50e5i)T^{2} \) |
| 71 | \( 1 + (138. - 37.0i)T + (3.09e5 - 1.78e5i)T^{2} \) |
| 73 | \( 1 + (-74.5 - 74.5i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 + 341.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-465. - 465. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + (514. + 137. i)T + (6.10e5 + 3.52e5i)T^{2} \) |
| 97 | \( 1 + (-437. - 1.63e3i)T + (-7.90e5 + 4.56e5i)T^{2} \) |
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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
−15.89031011031911433002306373158, −14.67560215968691626165186745636, −14.06050097784228173169020181827, −12.84603891168796184633416077542, −11.00317960943029454618146508331, −9.317144388850068006859453345184, −8.105107300794892507529793668781, −6.61100325351505645773137238541, −6.20709146787950855420480842993, −2.60282747841248606792642821834,
1.79835368688529991561865824219, 3.81806030469183697845215336569, 5.74472858046288672719141284757, 8.643014529656761444765432112378, 9.502458569252892394996607388972, 10.31059307452038702683674133724, 11.69211449527560494409547998235, 13.08950946820314946216958575075, 13.98187412347473574569747538471, 15.69662348099703045189056079903