| L(s) = 1 | + (0.245 + 1.39i)2-s + (−1.55 + 2.56i)3-s + (−1.87 + 0.684i)4-s + (1.32 + 0.233i)5-s + (−3.95 − 1.53i)6-s + (2.77 − 6.42i)7-s + (−1.41 − 2.44i)8-s + (−4.18 − 7.96i)9-s + 1.90i·10-s + (−1.44 − 8.20i)11-s + (1.16 − 5.88i)12-s + (−4.73 + 5.64i)13-s + (9.63 + 2.28i)14-s + (−2.65 + 3.04i)15-s + (3.06 − 2.57i)16-s − 22.9i·17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (−0.517 + 0.855i)3-s + (−0.469 + 0.171i)4-s + (0.265 + 0.0467i)5-s + (−0.659 − 0.255i)6-s + (0.396 − 0.918i)7-s + (−0.176 − 0.306i)8-s + (−0.464 − 0.885i)9-s + 0.190i·10-s + (−0.131 − 0.746i)11-s + (0.0967 − 0.490i)12-s + (−0.364 + 0.433i)13-s + (0.687 + 0.163i)14-s + (−0.177 + 0.202i)15-s + (0.191 − 0.160i)16-s − 1.35i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.354i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.935 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.05093 - 0.192475i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05093 - 0.192475i\) |
| \(L(2)\) |
|
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 | \( 1 + (1.55 - 2.56i)T \) |
| 7 | \( 1 + (-2.77 + 6.42i)T \) |
| good | 5 | \( 1 + (-1.32 - 0.233i)T + (23.4 + 8.55i)T^{2} \) |
| 11 | \( 1 + (1.44 + 8.20i)T + (-113. + 41.3i)T^{2} \) |
| 13 | \( 1 + (4.73 - 5.64i)T + (-29.3 - 166. i)T^{2} \) |
| 17 | \( 1 + 22.9iT - 289T^{2} \) |
| 19 | \( 1 + 29.3iT - 361T^{2} \) |
| 23 | \( 1 + (6.08 + 5.10i)T + (91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-3.35 + 2.81i)T + (146. - 828. i)T^{2} \) |
| 31 | \( 1 + (-16.9 - 46.5i)T + (-736. + 617. i)T^{2} \) |
| 37 | \( 1 + (-24.7 - 42.9i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-27.7 + 33.0i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-2.78 - 1.01i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (22.4 - 61.6i)T + (-1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 + (48.0 + 83.2i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-31.6 + 37.7i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (-11.9 + 32.8i)T + (-2.85e3 - 2.39e3i)T^{2} \) |
| 67 | \( 1 + (-13.1 + 74.4i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (25.4 - 44.1i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (40.5 + 23.4i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-16.8 - 95.2i)T + (-5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (85.8 + 102. i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + 132. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-23.1 + 63.5i)T + (-7.20e3 - 6.04e3i)T^{2} \) |
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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
−11.13584273980397388268366952756, −10.06964745523817244326870503794, −9.348925769907900181675469384487, −8.322189377637402503264802363374, −7.11188433328638672254465283993, −6.31327580368461604010865713207, −5.03778597080552466235880935960, −4.50377816377906178022537749416, −3.10308231120938947368239353726, −0.49677611516531182798499454509,
1.57908370401330849612560643499, 2.42458623605137505068608430794, 4.20078046964962008745887635485, 5.59967671418312883773786787624, 6.01376124665977862059992099471, 7.66162604199257651330839948945, 8.271797049803366363258435465901, 9.592082926167568595275525948563, 10.40914684266393639711610023385, 11.41505561082047799619777707799
