L(s) = 1 | + (1.53 − 1.28i)2-s + (0.694 − 3.93i)4-s + (−11.4 + 4.18i)5-s + (3.15 + 17.9i)7-s + (−4.00 − 6.92i)8-s + (−12.2 + 21.1i)10-s + (−63.8 − 23.2i)11-s + (−28.0 − 23.5i)13-s + (27.8 + 23.3i)14-s + (−15.0 − 5.47i)16-s + (−59.2 + 102. i)17-s + (51.9 + 89.9i)19-s + (8.49 + 48.2i)20-s + (−127. + 46.4i)22-s + (2.68 − 15.2i)23-s + ⋯ |
L(s) = 1 | + (0.541 − 0.454i)2-s + (0.0868 − 0.492i)4-s + (−1.02 + 0.374i)5-s + (0.170 + 0.966i)7-s + (−0.176 − 0.306i)8-s + (−0.386 + 0.670i)10-s + (−1.75 − 0.637i)11-s + (−0.599 − 0.502i)13-s + (0.531 + 0.446i)14-s + (−0.234 − 0.0855i)16-s + (−0.845 + 1.46i)17-s + (0.626 + 1.08i)19-s + (0.0950 + 0.538i)20-s + (−1.23 + 0.450i)22-s + (0.0242 − 0.137i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.670 - 0.741i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.670 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.166615 + 0.375476i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.166615 + 0.375476i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.53 + 1.28i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (11.4 - 4.18i)T + (95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (-3.15 - 17.9i)T + (-322. + 117. i)T^{2} \) |
| 11 | \( 1 + (63.8 + 23.2i)T + (1.01e3 + 855. i)T^{2} \) |
| 13 | \( 1 + (28.0 + 23.5i)T + (381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (59.2 - 102. i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-51.9 - 89.9i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-2.68 + 15.2i)T + (-1.14e4 - 4.16e3i)T^{2} \) |
| 29 | \( 1 + (-113. + 95.1i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-15.3 + 86.8i)T + (-2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (47.5 - 82.3i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (188. + 158. i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (94.8 + 34.5i)T + (6.09e4 + 5.11e4i)T^{2} \) |
| 47 | \( 1 + (-24.8 - 140. i)T + (-9.75e4 + 3.55e4i)T^{2} \) |
| 53 | \( 1 + 213.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-480. + 174. i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (75.9 + 431. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (680. + 570. i)T + (5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (202. - 350. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-37.3 - 64.6i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (469. - 393. i)T + (8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-165. + 138. i)T + (9.92e4 - 5.63e5i)T^{2} \) |
| 89 | \( 1 + (-416. - 721. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-1.13e3 - 412. i)T + (6.99e5 + 5.86e5i)T^{2} \) |
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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
−12.61187628190136755748305214231, −11.87012824928051600817396882796, −10.89365118763782276516024226307, −10.12337416606679218209379498219, −8.427561436351129778561381465108, −7.75775513174654055744451284681, −6.06830807036398786061116339043, −5.05542952087583451769502820645, −3.56783254273428420168623134326, −2.39903371176151743689617192541,
0.15065822496424915157315852772, 2.84224169577447337015869145162, 4.52493540465269923160739025113, 4.98569064018698999172781819067, 7.17012824859537516711997489222, 7.39396190715348664876593764660, 8.674366605262786181602002172231, 10.10484192044040853918697675260, 11.27012590378450213557977696727, 12.09470010352230886042533103044