| L(s) = 1 | + (1.44 + 0.832i)2-s + (−2.61 − 4.52i)4-s − 22.0i·8-s + (−25.4 + 14.6i)11-s + (−2.58 + 4.47i)16-s − 48.9·22-s + (−157. − 90.8i)23-s + (62.5 + 108. i)25-s + 304. i·29-s + (−160. + 92.3i)32-s + (5.29 − 9.16i)37-s − 534.·43-s + (133. + 76.7i)44-s + (−151. − 261. i)46-s + 208. i·50-s + ⋯ |
| L(s) = 1 | + (0.509 + 0.294i)2-s + (−0.326 − 0.566i)4-s − 0.973i·8-s + (−0.697 + 0.402i)11-s + (−0.0403 + 0.0699i)16-s − 0.473·22-s + (−1.42 − 0.823i)23-s + (0.5 + 0.866i)25-s + 1.94i·29-s + (−0.884 + 0.510i)32-s + (0.0235 − 0.0407i)37-s − 1.89·43-s + (0.455 + 0.263i)44-s + (−0.484 − 0.839i)46-s + 0.588i·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2359043355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2359043355\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + (-1.44 - 0.832i)T + (4 + 6.92i)T^{2} \) |
| 5 | \( 1 + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (25.4 - 14.6i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 2.19e3T^{2} \) |
| 17 | \( 1 + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (157. + 90.8i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 304. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-5.29 + 9.16i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 + 534.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (665. - 384. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (370 + 640. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 1.17e3iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-692 + 1.19e3i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 5.71e5T^{2} \) |
| 89 | \( 1 + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 9.12e5T^{2} \) |
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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
−10.91049296238306311799981128705, −10.24341312625436929152523588753, −9.384777037254632941616639439418, −8.374969621442176486037734477202, −7.22177286936302215410399084293, −6.31583837319218345178611587776, −5.28582279229079505295431326969, −4.55672414483622821714623369079, −3.27105642080885965585312187985, −1.61245338479461326529084267882,
0.06187976858386731759130845655, 2.17056808283481718442574377372, 3.30318803730018572194221817996, 4.31821222294357960359736849748, 5.32369735725816730750849000366, 6.38532243130747487671143289404, 7.88421954352548387295682239119, 8.209090591885607723526147407667, 9.470666359594375631651482239190, 10.37450918381794273722758359880
