| L(s) = 1 | + (−0.0274 − 1.41i)2-s + (−0.523 + 1.65i)3-s + (−1.99 + 0.0777i)4-s + (−0.720 − 0.552i)5-s + (2.34 + 0.694i)6-s + (−0.375 − 1.40i)7-s + (0.164 + 2.82i)8-s + (−2.45 − 1.72i)9-s + (−0.761 + 1.03i)10-s + (−4.59 − 0.605i)11-s + (0.917 − 3.34i)12-s + (−0.606 − 4.60i)13-s + (−1.97 + 0.569i)14-s + (1.28 − 0.900i)15-s + (3.98 − 0.310i)16-s + 2.96i·17-s + ⋯ |
| L(s) = 1 | + (−0.0194 − 0.999i)2-s + (−0.302 + 0.953i)3-s + (−0.999 + 0.0388i)4-s + (−0.322 − 0.247i)5-s + (0.958 + 0.283i)6-s + (−0.142 − 0.530i)7-s + (0.0583 + 0.998i)8-s + (−0.817 − 0.576i)9-s + (−0.240 + 0.326i)10-s + (−1.38 − 0.182i)11-s + (0.264 − 0.964i)12-s + (−0.168 − 1.27i)13-s + (−0.527 + 0.152i)14-s + (0.333 − 0.232i)15-s + (0.996 − 0.0777i)16-s + 0.720i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.102i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0140553 + 0.274787i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0140553 + 0.274787i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0274 + 1.41i)T \) |
| 3 | \( 1 + (0.523 - 1.65i)T \) |
| good | 5 | \( 1 + (0.720 + 0.552i)T + (1.29 + 4.82i)T^{2} \) |
| 7 | \( 1 + (0.375 + 1.40i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (4.59 + 0.605i)T + (10.6 + 2.84i)T^{2} \) |
| 13 | \( 1 + (0.606 + 4.60i)T + (-12.5 + 3.36i)T^{2} \) |
| 17 | \( 1 - 2.96iT - 17T^{2} \) |
| 19 | \( 1 + (6.72 + 2.78i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (1.45 - 5.43i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (3.06 + 3.99i)T + (-7.50 + 28.0i)T^{2} \) |
| 31 | \( 1 + (-4.46 - 7.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.69 + 0.702i)T + (26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (0.422 - 1.57i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-6.75 - 0.889i)T + (41.5 + 11.1i)T^{2} \) |
| 47 | \( 1 + (4.19 + 2.42i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.809 - 1.95i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (1.09 + 0.836i)T + (15.2 + 56.9i)T^{2} \) |
| 61 | \( 1 + (-0.841 - 1.09i)T + (-15.7 + 58.9i)T^{2} \) |
| 67 | \( 1 + (5.80 - 0.763i)T + (64.7 - 17.3i)T^{2} \) |
| 71 | \( 1 + (-8.61 + 8.61i)T - 71iT^{2} \) |
| 73 | \( 1 + (10.3 + 10.3i)T + 73iT^{2} \) |
| 79 | \( 1 + (-7.01 - 4.05i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (7.43 - 5.70i)T + (21.4 - 80.1i)T^{2} \) |
| 89 | \( 1 + (7.32 - 7.32i)T - 89iT^{2} \) |
| 97 | \( 1 + (-4.10 + 7.11i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.97729743615749240349123313202, −10.52383966931658423269068255484, −9.872370597383121407158212593598, −8.588539577029177551745102533672, −7.900054782555845040027439913427, −5.88408337224446956591708907182, −4.88070923741341829158053640406, −3.92118287466000643055760238554, −2.75770731588503707718446351523, −0.20424587999787477327303939075,
2.40704689473399233051554405578, 4.39888068877184667356388507242, 5.59987913419172204713944195744, 6.49386330730351889097742575902, 7.38602042902494830877926179834, 8.172353267261998567817986738822, 9.111658853142929033492413874715, 10.39753089053982043721467613211, 11.52376217174712314263235305835, 12.57674194671220791593006975540
