| L(s) = 1 | + (0.371 − 1.96i)2-s + (−1.67 − 0.448i)3-s + (−3.72 − 1.45i)4-s + (5.47 − 1.46i)5-s + (−1.50 + 3.12i)6-s + (−1.24 + 6.88i)7-s + (−4.25 + 6.77i)8-s + (2.59 + 1.50i)9-s + (−0.849 − 11.2i)10-s + (−16.2 − 4.36i)11-s + (5.57 + 4.11i)12-s + (−17.7 + 17.7i)13-s + (13.0 + 5.00i)14-s − 9.81·15-s + (11.7 + 10.8i)16-s + (4.91 − 2.83i)17-s + ⋯ |
| L(s) = 1 | + (0.185 − 0.982i)2-s + (−0.557 − 0.149i)3-s + (−0.931 − 0.364i)4-s + (1.09 − 0.293i)5-s + (−0.250 + 0.520i)6-s + (−0.177 + 0.984i)7-s + (−0.531 + 0.847i)8-s + (0.288 + 0.166i)9-s + (−0.0849 − 1.12i)10-s + (−1.48 − 0.396i)11-s + (0.464 + 0.342i)12-s + (−1.36 + 1.36i)13-s + (0.933 + 0.357i)14-s − 0.654·15-s + (0.733 + 0.679i)16-s + (0.289 − 0.166i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.455694 + 0.302788i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.455694 + 0.302788i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.371 + 1.96i)T \) |
| 3 | \( 1 + (1.67 + 0.448i)T \) |
| 7 | \( 1 + (1.24 - 6.88i)T \) |
| good | 5 | \( 1 + (-5.47 + 1.46i)T + (21.6 - 12.5i)T^{2} \) |
| 11 | \( 1 + (16.2 + 4.36i)T + (104. + 60.5i)T^{2} \) |
| 13 | \( 1 + (17.7 - 17.7i)T - 169iT^{2} \) |
| 17 | \( 1 + (-4.91 + 2.83i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-3.96 - 14.7i)T + (-312. + 180.5i)T^{2} \) |
| 23 | \( 1 + (29.4 + 17.0i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-18.7 - 18.7i)T + 841iT^{2} \) |
| 31 | \( 1 + (12.9 - 7.50i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-10.7 - 40.1i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 - 73.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-21.1 + 21.1i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (23.1 + 13.3i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (66.9 + 17.9i)T + (2.43e3 + 1.40e3i)T^{2} \) |
| 59 | \( 1 + (18.9 - 70.8i)T + (-3.01e3 - 1.74e3i)T^{2} \) |
| 61 | \( 1 + (16.4 + 61.4i)T + (-3.22e3 + 1.86e3i)T^{2} \) |
| 67 | \( 1 + (9.66 - 36.0i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 - 4.09iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (66.4 + 115. i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (20.4 - 35.3i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (25.0 - 25.0i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (2.52 - 4.38i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 29.6iT - 9.40e3T^{2} \) |
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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
−11.71190326345507838240097317414, −10.49846974134554267394427169702, −9.840129920206871296078988316922, −9.118674702023965943908711473083, −7.919508402141457215067103163522, −6.20513968884217283551791785667, −5.45334352355789208977879211205, −4.66359904915433771988261464007, −2.70739713540560385082127908916, −1.85813844365798037150463676358,
0.24043176443045805397162878294, 2.76617198856635098781485874551, 4.46170486137045848844310261662, 5.43407849960647289438014924508, 6.08817799921183752704255940403, 7.46485733012939119052468574945, 7.76019890434894865173889546498, 9.745258161428232283483758038383, 9.901514229333444963170877974156, 10.85446218725749106212467261607
