L(s) = 1 | + 1.76i·3-s + 4.62i·7-s − 0.103·9-s − 5.52·11-s + 5.49i·13-s + 6.62i·17-s + 19-s − 8.14·21-s + 4.14i·23-s + 5.10i·27-s + 7.87·29-s + 1.25·31-s − 9.72i·33-s + 0.387i·37-s − 9.67·39-s + ⋯ |
L(s) = 1 | + 1.01i·3-s + 1.74i·7-s − 0.0343·9-s − 1.66·11-s + 1.52i·13-s + 1.60i·17-s + 0.229·19-s − 1.77·21-s + 0.865i·23-s + 0.982i·27-s + 1.46·29-s + 0.224·31-s − 1.69i·33-s + 0.0637i·37-s − 1.54·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.530556371\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.530556371\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 1.76iT - 3T^{2} \) |
| 7 | \( 1 - 4.62iT - 7T^{2} \) |
| 11 | \( 1 + 5.52T + 11T^{2} \) |
| 13 | \( 1 - 5.49iT - 13T^{2} \) |
| 17 | \( 1 - 6.62iT - 17T^{2} \) |
| 23 | \( 1 - 4.14iT - 23T^{2} \) |
| 29 | \( 1 - 7.87T + 29T^{2} \) |
| 31 | \( 1 - 1.25T + 31T^{2} \) |
| 37 | \( 1 - 0.387iT - 37T^{2} \) |
| 41 | \( 1 - 6.77T + 41T^{2} \) |
| 43 | \( 1 + 10.9iT - 43T^{2} \) |
| 47 | \( 1 + 1.72iT - 47T^{2} \) |
| 53 | \( 1 - 1.49iT - 53T^{2} \) |
| 59 | \( 1 + 0.626T + 59T^{2} \) |
| 61 | \( 1 - 15.0T + 61T^{2} \) |
| 67 | \( 1 + 5.22iT - 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 + 4.83iT - 73T^{2} \) |
| 79 | \( 1 - 2.98T + 79T^{2} \) |
| 83 | \( 1 + 2.74iT - 83T^{2} \) |
| 89 | \( 1 + 4.27T + 89T^{2} \) |
| 97 | \( 1 - 13.7iT - 97T^{2} \) |
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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
−8.870331992407754769187663822878, −8.479531534791128939491059187719, −7.58999428143204628278591229663, −6.53895679688382680505264045220, −5.73310335969911330593803927912, −5.18080520841151886140763018977, −4.45010469299731146659966239112, −3.55775146505649939942628148591, −2.55965775432027961233084766395, −1.82820213190766681649474878966,
0.54236866150373580835468853725, 0.949178259634359153352993798689, 2.56754261758885739691919033778, 3.05364215897853056653497623342, 4.40923591793889645791041773897, 4.97342780122875455047696196600, 5.95484426972351864196793547063, 6.91259796970260692414781318447, 7.36718153166328054233246870416, 7.889893680909231456890231369774