| L(s) = 1 | + (0.705 − 0.471i)3-s + (−6.65 − 1.32i)5-s + (−11.2 + 2.23i)7-s + (−3.16 + 7.64i)9-s + (1.14 − 1.71i)11-s + (−0.784 − 0.784i)13-s + (−5.32 + 2.20i)15-s + (15.3 + 7.33i)17-s + (−8.11 − 19.5i)19-s + (−6.86 + 6.86i)21-s + (−24.3 − 16.2i)23-s + (19.4 + 8.06i)25-s + (2.86 + 14.3i)27-s + (−7.57 + 38.0i)29-s + (−23.9 − 35.8i)31-s + ⋯ |
| L(s) = 1 | + (0.235 − 0.157i)3-s + (−1.33 − 0.264i)5-s + (−1.60 + 0.318i)7-s + (−0.352 + 0.849i)9-s + (0.103 − 0.155i)11-s + (−0.0603 − 0.0603i)13-s + (−0.354 + 0.146i)15-s + (0.902 + 0.431i)17-s + (−0.426 − 1.03i)19-s + (−0.327 + 0.327i)21-s + (−1.05 − 0.707i)23-s + (0.779 + 0.322i)25-s + (0.105 + 0.532i)27-s + (−0.261 + 1.31i)29-s + (−0.773 − 1.15i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00733873 + 0.0710732i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00733873 + 0.0710732i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (-15.3 - 7.33i)T \) |
| good | 3 | \( 1 + (-0.705 + 0.471i)T + (3.44 - 8.31i)T^{2} \) |
| 5 | \( 1 + (6.65 + 1.32i)T + (23.0 + 9.56i)T^{2} \) |
| 7 | \( 1 + (11.2 - 2.23i)T + (45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (-1.14 + 1.71i)T + (-46.3 - 111. i)T^{2} \) |
| 13 | \( 1 + (0.784 + 0.784i)T + 169iT^{2} \) |
| 19 | \( 1 + (8.11 + 19.5i)T + (-255. + 255. i)T^{2} \) |
| 23 | \( 1 + (24.3 + 16.2i)T + (202. + 488. i)T^{2} \) |
| 29 | \( 1 + (7.57 - 38.0i)T + (-776. - 321. i)T^{2} \) |
| 31 | \( 1 + (23.9 + 35.8i)T + (-367. + 887. i)T^{2} \) |
| 37 | \( 1 + (-45.2 + 30.2i)T + (523. - 1.26e3i)T^{2} \) |
| 41 | \( 1 + (67.0 - 13.3i)T + (1.55e3 - 643. i)T^{2} \) |
| 43 | \( 1 + (3.43 - 8.30i)T + (-1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-40.2 - 40.2i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-9.59 - 23.1i)T + (-1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (12.9 + 5.36i)T + (2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-7.65 - 38.4i)T + (-3.43e3 + 1.42e3i)T^{2} \) |
| 67 | \( 1 - 41.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (79.4 - 53.0i)T + (1.92e3 - 4.65e3i)T^{2} \) |
| 73 | \( 1 + (40.8 + 8.12i)T + (4.92e3 + 2.03e3i)T^{2} \) |
| 79 | \( 1 + (6.76 - 10.1i)T + (-2.38e3 - 5.76e3i)T^{2} \) |
| 83 | \( 1 + (20.2 - 8.38i)T + (4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (56.0 - 56.0i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (-19.7 + 99.4i)T + (-8.69e3 - 3.60e3i)T^{2} \) |
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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
−13.16353348168950417295993287534, −12.56453108489373243714432797981, −11.54500712922383480767197624043, −10.41865498703455162792531746371, −9.139355627365157122245987742871, −8.160300459395513040763213624013, −7.15440887692669676081827384341, −5.78900134368597082718325861599, −4.10563099120265241439712731079, −2.87481745835692926074835844951,
0.04430321039069886176169711827, 3.31644252862560598523509290925, 3.88120651853539438951317380123, 6.02350094344745803950302741518, 7.11165177682328144009655285622, 8.175151422246945536547631731173, 9.507491509268153374373800546472, 10.25468313682374985156170679454, 11.82977593729105643355380836244, 12.21689977230360834503328145020
