| L(s) = 1 | + (−1.23 − 0.682i)2-s + (0.504 − 0.694i)3-s + (1.06 + 1.69i)4-s + (−2.09 − 0.771i)5-s + (−1.09 + 0.516i)6-s + (2.12 + 2.12i)7-s + (−0.172 − 2.82i)8-s + (0.699 + 2.15i)9-s + (2.07 + 2.38i)10-s + (−3.14 + 1.60i)11-s + (1.71 + 0.110i)12-s + (0.650 + 2.00i)13-s + (−1.18 − 4.08i)14-s + (−1.59 + 1.06i)15-s + (−1.71 + 3.61i)16-s + (2.76 + 0.437i)17-s + ⋯ |
| L(s) = 1 | + (−0.875 − 0.482i)2-s + (0.291 − 0.401i)3-s + (0.534 + 0.845i)4-s + (−0.938 − 0.345i)5-s + (−0.448 + 0.210i)6-s + (0.802 + 0.802i)7-s + (−0.0608 − 0.998i)8-s + (0.233 + 0.717i)9-s + (0.655 + 0.754i)10-s + (−0.946 + 0.482i)11-s + (0.494 + 0.0317i)12-s + (0.180 + 0.555i)13-s + (−0.316 − 1.09i)14-s + (−0.411 + 0.275i)15-s + (−0.428 + 0.903i)16-s + (0.669 + 0.106i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.425i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.829406 + 0.185263i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.829406 + 0.185263i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 + 0.682i)T \) |
| 5 | \( 1 + (2.09 + 0.771i)T \) |
| good | 3 | \( 1 + (-0.504 + 0.694i)T + (-0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 + (-2.12 - 2.12i)T + 7iT^{2} \) |
| 11 | \( 1 + (3.14 - 1.60i)T + (6.46 - 8.89i)T^{2} \) |
| 13 | \( 1 + (-0.650 - 2.00i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.76 - 0.437i)T + (16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-0.887 - 0.140i)T + (18.0 + 5.87i)T^{2} \) |
| 23 | \( 1 + (-1.53 - 3.01i)T + (-13.5 + 18.6i)T^{2} \) |
| 29 | \( 1 + (5.07 - 0.804i)T + (27.5 - 8.96i)T^{2} \) |
| 31 | \( 1 + (-0.567 - 0.781i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (0.787 + 2.42i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-5.18 + 1.68i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 6.24T + 43T^{2} \) |
| 47 | \( 1 + (-9.20 + 1.45i)T + (44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (8.11 - 11.1i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-8.06 - 4.11i)T + (34.6 + 47.7i)T^{2} \) |
| 61 | \( 1 + (10.8 - 5.52i)T + (35.8 - 49.3i)T^{2} \) |
| 67 | \( 1 + (-10.2 + 7.42i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (8.07 + 5.86i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (4.48 - 2.28i)T + (42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (5.32 + 3.86i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (0.311 + 0.428i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (3.43 - 10.5i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (1.89 + 11.9i)T + (-92.2 + 29.9i)T^{2} \) |
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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
−11.23365063452925132436776219682, −10.61656934592023474355363085103, −9.301813392316488145934893753295, −8.554167671047484687347264577709, −7.61682505026086375851479965177, −7.42087199838401437064660472105, −5.47593319428755792443774484926, −4.25909710083098703101994890963, −2.74202756373938863450831589931, −1.57348097836889578709475880334,
0.77508142757468679379136179478, 2.97423124337426577566805963482, 4.25738517946830392986916168636, 5.50632582595945990826913981855, 6.84172464477366552145284519584, 7.77016324480805186570316002485, 8.188247619276198983681975012534, 9.322613762891198505346343649518, 10.39224971399368667283448518071, 10.87167462946809294511008550877
