| L(s) = 1 | + (−1.53 − 1.28i)2-s + (0.694 + 3.93i)4-s + (−3.45 − 1.25i)5-s + (0.157 − 0.890i)7-s + (4.00 − 6.92i)8-s + (3.67 + 6.35i)10-s + (0.906 − 0.329i)11-s + (−34.0 + 28.5i)13-s + (−1.38 + 1.16i)14-s + (−15.0 + 5.47i)16-s + (23.7 + 41.1i)17-s + (−40.7 + 70.5i)19-s + (2.55 − 14.4i)20-s + (−1.81 − 0.659i)22-s + (22.7 + 129. i)23-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (0.0868 + 0.492i)4-s + (−0.308 − 0.112i)5-s + (0.00847 − 0.0480i)7-s + (0.176 − 0.306i)8-s + (0.116 + 0.201i)10-s + (0.0248 − 0.00904i)11-s + (−0.725 + 0.608i)13-s + (−0.0264 + 0.0221i)14-s + (−0.234 + 0.0855i)16-s + (0.339 + 0.587i)17-s + (−0.491 + 0.851i)19-s + (0.0285 − 0.161i)20-s + (−0.0175 − 0.00639i)22-s + (0.206 + 1.17i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.141 - 0.989i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.141 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.540514 + 0.468708i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.540514 + 0.468708i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (3.45 + 1.25i)T + (95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (-0.157 + 0.890i)T + (-322. - 117. i)T^{2} \) |
| 11 | \( 1 + (-0.906 + 0.329i)T + (1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (34.0 - 28.5i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-23.7 - 41.1i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (40.7 - 70.5i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-22.7 - 129. i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (-108. - 90.9i)T + (4.23e3 + 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-33.1 - 188. i)T + (-2.79e4 + 1.01e4i)T^{2} \) |
| 37 | \( 1 + (-172. - 299. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (268. - 224. i)T + (1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (51.5 - 18.7i)T + (6.09e4 - 5.11e4i)T^{2} \) |
| 47 | \( 1 + (-65.7 + 373. i)T + (-9.75e4 - 3.55e4i)T^{2} \) |
| 53 | \( 1 + 583.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (180. + 65.5i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-94.1 + 533. i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (543. - 455. i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (484. + 839. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-253. + 439. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (716. + 601. i)T + (8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-704. - 590. i)T + (9.92e4 + 5.63e5i)T^{2} \) |
| 89 | \( 1 + (17.2 - 29.8i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-509. + 185. i)T + (6.99e5 - 5.86e5i)T^{2} \) |
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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
−12.28088480120750034391734197360, −11.74400183723466029456708890151, −10.49962052386040350401334519977, −9.712311647932179340880215379918, −8.537318173267451377603889926307, −7.64914500985367036198032193473, −6.38668835082520240082694576382, −4.71393407882646442528509121462, −3.34352282900469637924968216175, −1.61665764511847706400340080713,
0.40255517450608571316085290627, 2.58986738838274329690044361966, 4.48335112941971969114567268532, 5.80067028886222032931233554321, 7.07444462193775457394759476835, 7.924624364708113382773944443182, 9.054561990170384071161263995859, 10.04183416310867980549010659834, 11.04196477601001072685265198475, 12.08199500533504667024028315196
