| L(s) = 1 | + (0.182 + 1.40i)2-s + (1.93 − 0.385i)3-s + (−1.93 + 0.511i)4-s + (−0.787 − 1.17i)5-s + (0.893 + 2.64i)6-s + (−2.16 − 0.896i)7-s + (−1.06 − 2.61i)8-s + (0.830 − 0.343i)9-s + (1.50 − 1.31i)10-s + (−1.08 + 5.44i)11-s + (−3.54 + 1.73i)12-s + (1.49 − 2.24i)13-s + (0.862 − 3.19i)14-s + (−1.97 − 1.97i)15-s + (3.47 − 1.97i)16-s + (3.43 − 3.43i)17-s + ⋯ |
| L(s) = 1 | + (0.128 + 0.991i)2-s + (1.11 − 0.222i)3-s + (−0.966 + 0.255i)4-s + (−0.352 − 0.527i)5-s + (0.364 + 1.08i)6-s + (−0.818 − 0.338i)7-s + (−0.378 − 0.925i)8-s + (0.276 − 0.114i)9-s + (0.477 − 0.417i)10-s + (−0.326 + 1.64i)11-s + (−1.02 + 0.500i)12-s + (0.415 − 0.621i)13-s + (0.230 − 0.855i)14-s + (−0.511 − 0.511i)15-s + (0.869 − 0.494i)16-s + (0.834 − 0.834i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.641 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.641 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.945890 + 0.441848i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.945890 + 0.441848i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.182 - 1.40i)T \) |
| good | 3 | \( 1 + (-1.93 + 0.385i)T + (2.77 - 1.14i)T^{2} \) |
| 5 | \( 1 + (0.787 + 1.17i)T + (-1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (2.16 + 0.896i)T + (4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (1.08 - 5.44i)T + (-10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (-1.49 + 2.24i)T + (-4.97 - 12.0i)T^{2} \) |
| 17 | \( 1 + (-3.43 + 3.43i)T - 17iT^{2} \) |
| 19 | \( 1 + (1.24 + 0.828i)T + (7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (-2.14 - 5.17i)T + (-16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (-1.63 - 8.20i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + 5.17iT - 31T^{2} \) |
| 37 | \( 1 + (6.79 - 4.53i)T + (14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (2.24 + 5.42i)T + (-28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (-4.16 - 0.827i)T + (39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (0.733 - 0.733i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.575 + 2.89i)T + (-48.9 - 20.2i)T^{2} \) |
| 59 | \( 1 + (3.31 + 4.96i)T + (-22.5 + 54.5i)T^{2} \) |
| 61 | \( 1 + (-0.382 + 0.0761i)T + (56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + (-1.67 + 0.333i)T + (61.8 - 25.6i)T^{2} \) |
| 71 | \( 1 + (0.843 + 0.349i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-11.9 + 4.96i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (5.30 + 5.30i)T + 79iT^{2} \) |
| 83 | \( 1 + (1.28 + 0.860i)T + (31.7 + 76.6i)T^{2} \) |
| 89 | \( 1 + (3.98 - 9.63i)T + (-62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 - 4.23iT - 97T^{2} \) |
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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
−15.16282634014120635430297967927, −14.09595802325133143598164839363, −13.12398037869686224039663038729, −12.39904241020006253558743075990, −9.990264272673399080274705453261, −9.032393281009843726666561295504, −7.86898707252461900400854992443, −7.02280313051572365180808537025, −5.06941086432272467173660470968, −3.40760729249051938312299483831,
2.88704193594896877203566936409, 3.71147529266525579458612902206, 6.02953555304155005751479377059, 8.267333134549782351765104272540, 8.990131419388331682439164602926, 10.28462562343082768192574374433, 11.32466443737539870285505589272, 12.65315049952064568301678984313, 13.76173294249095057115645552149, 14.44772256205723601909856344279
