| L(s) = 1 | − 3-s + 9-s − 4·11-s − 4i·13-s + 4i·17-s − 2i·19-s + 6i·23-s + 5·25-s − 27-s + 2i·31-s + 4·33-s + (−1 + 6i)37-s + 4i·39-s + 10·41-s − 2i·43-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.333·9-s − 1.20·11-s − 1.10i·13-s + 0.970i·17-s − 0.458i·19-s + 1.25i·23-s + 25-s − 0.192·27-s + 0.359i·31-s + 0.696·33-s + (−0.164 + 0.986i)37-s + 0.640i·39-s + 1.56·41-s − 0.304i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9058705526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9058705526\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 37 | \( 1 + (1 - 6i)T \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 2iT - 31T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 10iT - 59T^{2} \) |
| 61 | \( 1 - 4iT - 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 - 10iT - 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 - 12iT - 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−9.611726185419203424424481449544, −8.551445514778962421735366899732, −7.87008573367394174641397090543, −7.14055094203928302010126036451, −6.12138293160234634007709136847, −5.41034937911081408641758833854, −4.76456139651923326700383541313, −3.52503966620544229290575075582, −2.58591039505044380474303150722, −1.11489168612621006083652956919,
0.41251663879646985283880707533, 2.01336282264512785906863975052, 3.03779815729312474189601271033, 4.38511045596947263162602088329, 4.94888956751561572608561797321, 5.89619881402043408173709340186, 6.71881838342451378618128158020, 7.44194495628055465117252546551, 8.293837326944815249183565026568, 9.198957374390445584255459839649
