| L(s) = 1 | + (−0.245 − 1.39i)2-s + (−2.88 + 0.836i)3-s + (−1.87 + 0.684i)4-s + (−2.78 − 0.491i)5-s + (1.87 + 3.80i)6-s + (−2.31 − 6.60i)7-s + (1.41 + 2.44i)8-s + (7.60 − 4.81i)9-s + 4.00i·10-s + (−3.52 − 20.0i)11-s + (4.84 − 3.54i)12-s + (−2.95 + 3.51i)13-s + (−8.63 + 4.84i)14-s + (8.43 − 0.914i)15-s + (3.06 − 2.57i)16-s + 16.8i·17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (−0.960 + 0.278i)3-s + (−0.469 + 0.171i)4-s + (−0.557 − 0.0982i)5-s + (0.312 + 0.634i)6-s + (−0.330 − 0.943i)7-s + (0.176 + 0.306i)8-s + (0.844 − 0.535i)9-s + 0.400i·10-s + (−0.320 − 1.81i)11-s + (0.403 − 0.295i)12-s + (−0.226 + 0.270i)13-s + (−0.616 + 0.346i)14-s + (0.562 − 0.0609i)15-s + (0.191 − 0.160i)16-s + 0.993i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.276 - 0.961i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.276 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.183744 + 0.138334i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.183744 + 0.138334i\) |
| \(L(2)\) |
|
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 | \( 1 + (2.88 - 0.836i)T \) |
| 7 | \( 1 + (2.31 + 6.60i)T \) |
| good | 5 | \( 1 + (2.78 + 0.491i)T + (23.4 + 8.55i)T^{2} \) |
| 11 | \( 1 + (3.52 + 20.0i)T + (-113. + 41.3i)T^{2} \) |
| 13 | \( 1 + (2.95 - 3.51i)T + (-29.3 - 166. i)T^{2} \) |
| 17 | \( 1 - 16.8iT - 289T^{2} \) |
| 19 | \( 1 - 18.3iT - 361T^{2} \) |
| 23 | \( 1 + (-5.71 - 4.79i)T + (91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (23.3 - 19.5i)T + (146. - 828. i)T^{2} \) |
| 31 | \( 1 + (-7.02 - 19.2i)T + (-736. + 617. i)T^{2} \) |
| 37 | \( 1 + (-25.4 - 44.1i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-8.88 + 10.5i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-1.46 - 0.532i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-13.3 + 36.5i)T + (-1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 + (8.13 + 14.0i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-45.3 + 54.0i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (23.7 - 65.1i)T + (-2.85e3 - 2.39e3i)T^{2} \) |
| 67 | \( 1 + (17.0 - 96.9i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-13.8 + 24.0i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (46.1 + 26.6i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-10.1 - 57.5i)T + (-5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-43.8 - 52.2i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + 36.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (33.1 - 91.0i)T + (-7.20e3 - 6.04e3i)T^{2} \) |
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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
−11.23437357003554711818502373594, −10.59174786786130198947518299753, −9.909075355184375924867239822233, −8.636567854254729834796112961434, −7.69821092088392277595601594674, −6.42555890642433135457622542138, −5.44112399218809140303996092927, −4.08488366105267213400055695997, −3.44855444255640468061156876261, −1.13082107748845579862368594885,
0.14212927441960619761417902438, 2.33287310285754673711331860110, 4.39047197349593845551693938060, 5.18545670446446220076429981029, 6.19401919090021018972414389778, 7.28949407566596850134353073585, 7.67393490558240253824635925651, 9.275552557245988110525964768320, 9.835642464984051643863692188509, 11.10431437643235746119798549545
