| L(s) = 1 | + (0.774 + 1.18i)2-s + (−1.33 + 1.10i)3-s + (−0.800 + 1.83i)4-s + (−0.168 + 0.574i)5-s + (−2.33 − 0.726i)6-s + (2.73 + 2.36i)7-s + (−2.78 + 0.472i)8-s + (0.567 − 2.94i)9-s + (−0.810 + 0.245i)10-s + (−2.81 + 1.81i)11-s + (−0.952 − 3.33i)12-s + (−2.47 − 2.85i)13-s + (−0.685 + 5.06i)14-s + (−0.408 − 0.953i)15-s + (−2.71 − 2.93i)16-s + (1.55 + 0.708i)17-s + ⋯ |
| L(s) = 1 | + (0.547 + 0.836i)2-s + (−0.771 + 0.636i)3-s + (−0.400 + 0.916i)4-s + (−0.0754 + 0.257i)5-s + (−0.955 − 0.296i)6-s + (1.03 + 0.894i)7-s + (−0.985 + 0.166i)8-s + (0.189 − 0.981i)9-s + (−0.256 + 0.0776i)10-s + (−0.849 + 0.546i)11-s + (−0.274 − 0.961i)12-s + (−0.685 − 0.790i)13-s + (−0.183 + 1.35i)14-s + (−0.105 − 0.246i)15-s + (−0.679 − 0.733i)16-s + (0.376 + 0.171i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.216i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.126924 + 1.16034i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.126924 + 1.16034i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.774 - 1.18i)T \) |
| 3 | \( 1 + (1.33 - 1.10i)T \) |
| 23 | \( 1 + (3.51 - 3.26i)T \) |
| good | 5 | \( 1 + (0.168 - 0.574i)T + (-4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (-2.73 - 2.36i)T + (0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (2.81 - 1.81i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (2.47 + 2.85i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.55 - 0.708i)T + (11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.386 + 0.176i)T + (12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.0545 - 0.0249i)T + (18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-7.54 - 1.08i)T + (29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-9.04 + 2.65i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (2.14 - 7.29i)T + (-34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (3.74 - 0.537i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 8.85T + 47T^{2} \) |
| 53 | \( 1 + (-2.10 - 1.82i)T + (7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (3.59 + 4.14i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (1.40 - 9.79i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-4.30 + 6.69i)T + (-27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (11.2 + 7.21i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (0.421 + 0.923i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (9.95 - 8.63i)T + (11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-8.26 + 2.42i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-14.9 + 2.14i)T + (85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-6.21 - 1.82i)T + (81.6 + 52.4i)T^{2} \) |
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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
−12.21097368797117182999115969827, −11.65037840250625881515373687702, −10.49196201064481809996541228306, −9.463441390459863092516389764730, −8.217798357118339526869614782959, −7.42757260741525868246114726175, −6.02972904743635468466295131013, −5.24116394997226347307021181409, −4.53880919053541275664030350548, −2.88849639669178026949443498408,
0.858418609191364626166611941106, 2.37633392962548583050740728429, 4.36999338252314653310653802737, 5.02537261386450512520143401047, 6.22435613259409588682776975874, 7.47523528924239877586013765828, 8.475547855900031415314261397465, 10.10202299361897285560386513195, 10.69554925266410885443273442030, 11.61998099921368245325856566694
