| L(s) = 1 | + (−0.245 − 1.39i)2-s + (0.411 − 2.97i)3-s + (−1.87 + 0.684i)4-s + (6.35 + 1.12i)5-s + (−4.23 + 0.157i)6-s + (−0.735 + 6.96i)7-s + (1.41 + 2.44i)8-s + (−8.66 − 2.44i)9-s − 9.13i·10-s + (0.390 + 2.21i)11-s + (1.25 + 5.86i)12-s + (15.1 − 18.0i)13-s + (9.87 − 0.685i)14-s + (5.94 − 18.4i)15-s + (3.06 − 2.57i)16-s − 21.4i·17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (0.137 − 0.990i)3-s + (−0.469 + 0.171i)4-s + (1.27 + 0.224i)5-s + (−0.706 + 0.0261i)6-s + (−0.105 + 0.994i)7-s + (0.176 + 0.306i)8-s + (−0.962 − 0.271i)9-s − 0.913i·10-s + (0.0355 + 0.201i)11-s + (0.104 + 0.488i)12-s + (1.16 − 1.38i)13-s + (0.705 − 0.0489i)14-s + (0.396 − 1.22i)15-s + (0.191 − 0.160i)16-s − 1.25i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.400 + 0.916i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.04667 - 1.59936i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04667 - 1.59936i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.245 + 1.39i)T \) |
| 3 | \( 1 + (-0.411 + 2.97i)T \) |
| 7 | \( 1 + (0.735 - 6.96i)T \) |
| good | 5 | \( 1 + (-6.35 - 1.12i)T + (23.4 + 8.55i)T^{2} \) |
| 11 | \( 1 + (-0.390 - 2.21i)T + (-113. + 41.3i)T^{2} \) |
| 13 | \( 1 + (-15.1 + 18.0i)T + (-29.3 - 166. i)T^{2} \) |
| 17 | \( 1 + 21.4iT - 289T^{2} \) |
| 19 | \( 1 + 27.8iT - 361T^{2} \) |
| 23 | \( 1 + (-12.8 - 10.7i)T + (91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-2.17 + 1.82i)T + (146. - 828. i)T^{2} \) |
| 31 | \( 1 + (5.69 + 15.6i)T + (-736. + 617. i)T^{2} \) |
| 37 | \( 1 + (-20.4 - 35.4i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (4.45 - 5.30i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-58.2 - 21.1i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-20.0 + 55.2i)T + (-1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 + (35.4 + 61.4i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (11.2 - 13.3i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (-2.64 + 7.27i)T + (-2.85e3 - 2.39e3i)T^{2} \) |
| 67 | \( 1 + (20.7 - 117. i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (47.9 - 83.0i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-44.3 - 25.6i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-20.2 - 115. i)T + (-5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (31.7 + 37.8i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 - 10.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-8.44 + 23.2i)T + (-7.20e3 - 6.04e3i)T^{2} \) |
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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
−11.07600826628269877889035518757, −9.824460601264979084795860824364, −9.109904070880905458979401896281, −8.300100690656475244668333916702, −7.01987960734643013173199682201, −5.93530160019653352476365369204, −5.24953370130237794249160949562, −2.99763368516620695471053718372, −2.40063326545103168163197536669, −0.972404945816295619366839631033,
1.58413908081126255157670243941, 3.68631034404315389412375180566, 4.49187899060604029164302996394, 5.89759668241451083499690269206, 6.30092141878705675419199457653, 7.82518882259149781120025550374, 8.933505008969239004275376946136, 9.406081911842820774688822088907, 10.51530979965640045045073740851, 10.84131597916136907607673859273
