| L(s) = 1 | + (0.0599 − 1.73i)3-s + (0.392 − 2.22i)5-s + (3.55 − 2.98i)7-s + (−2.99 − 0.207i)9-s + (−0.0857 − 0.486i)11-s + (−1.29 + 0.472i)13-s + (−3.83 − 0.813i)15-s + (2.34 − 4.06i)17-s + (0.368 + 0.638i)19-s + (−4.95 − 6.33i)21-s + (3.71 + 3.11i)23-s + (−0.110 − 0.0403i)25-s + (−0.538 + 5.16i)27-s + (−6.04 − 2.20i)29-s + (7.95 + 6.67i)31-s + ⋯ |
| L(s) = 1 | + (0.0346 − 0.999i)3-s + (0.175 − 0.996i)5-s + (1.34 − 1.12i)7-s + (−0.997 − 0.0692i)9-s + (−0.0258 − 0.146i)11-s + (−0.359 + 0.130i)13-s + (−0.989 − 0.210i)15-s + (0.569 − 0.986i)17-s + (0.0845 + 0.146i)19-s + (−1.08 − 1.38i)21-s + (0.775 + 0.650i)23-s + (−0.0221 − 0.00806i)25-s + (−0.103 + 0.994i)27-s + (−1.12 − 0.408i)29-s + (1.42 + 1.19i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.677 + 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.706584 - 1.61039i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.706584 - 1.61039i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.0599 + 1.73i)T \) |
| good | 5 | \( 1 + (-0.392 + 2.22i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-3.55 + 2.98i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (0.0857 + 0.486i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (1.29 - 0.472i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.34 + 4.06i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.368 - 0.638i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.71 - 3.11i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (6.04 + 2.20i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-7.95 - 6.67i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (3.34 - 5.78i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (6.29 - 2.29i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.00111 - 0.00633i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (7.24 - 6.07i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 4.89T + 53T^{2} \) |
| 59 | \( 1 + (-1.96 + 11.1i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (5.09 - 4.27i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-6.31 + 2.29i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-4.36 + 7.56i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (8.06 + 13.9i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.82 - 1.39i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (1.25 + 0.457i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (1.21 + 2.10i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.94 - 11.0i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−9.773682316293345202054557826804, −8.798400327360641168106830520694, −8.016824466371636956441604526945, −7.47027285947055775594125038230, −6.58071224674009592335489607182, −5.12909822393308436244596950257, −4.86524367431116689825338598331, −3.27832926314580630717191970588, −1.67140493348959483834556274210, −0.919895775729456735691678653088,
2.09874312466581953512443688432, 3.00079525455897782542268428416, 4.23337838095893640537479393666, 5.24734882140331717733724603190, 5.84153737131263894802620679786, 7.06577064282679211556549383239, 8.222687655299693042757360322429, 8.711199105630555175595339728632, 9.786744737388009555209767631138, 10.47766907807941773130426623506
