| L(s) = 1 | + 1.73i·3-s − 7.06i·7-s − 2.99·9-s − 1.70i·11-s + 19.3·13-s + 13.0·17-s + 13.2i·19-s + 12.2·21-s − 5.26i·23-s − 5.19i·27-s − 33.0·29-s − 60.5i·31-s + 2.94·33-s − 46.9·37-s + 33.5i·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.00i·7-s − 0.333·9-s − 0.154i·11-s + 1.49·13-s + 0.765·17-s + 0.696i·19-s + 0.582·21-s − 0.228i·23-s − 0.192i·27-s − 1.13·29-s − 1.95i·31-s + 0.0893·33-s − 1.26·37-s + 0.860i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.001983495\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.001983495\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 7.06iT - 49T^{2} \) |
| 11 | \( 1 + 1.70iT - 121T^{2} \) |
| 13 | \( 1 - 19.3T + 169T^{2} \) |
| 17 | \( 1 - 13.0T + 289T^{2} \) |
| 19 | \( 1 - 13.2iT - 361T^{2} \) |
| 23 | \( 1 + 5.26iT - 529T^{2} \) |
| 29 | \( 1 + 33.0T + 841T^{2} \) |
| 31 | \( 1 + 60.5iT - 961T^{2} \) |
| 37 | \( 1 + 46.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 65.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 20.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 40.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 11.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 39.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 32.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 2.76iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 67.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 81.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 143. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 126. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 11.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + 49.5T + 9.40e3T^{2} \) |
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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
−8.742951214800734771466546180985, −7.86701240860452898400288810173, −7.34312437729976130018953495826, −6.06696285361518847287681048364, −5.77634729156601719488662180469, −4.44750040954758751317619385886, −3.87409951348381368450519811962, −3.17050193058151628214619706588, −1.67377402850498233373253276306, −0.55819606189184190327117983457,
1.04420744916443635974343124701, 2.00061084691006413130572957361, 3.05553019783132899656916866714, 3.87287378445777130441097759928, 5.27436099688285957136776948949, 5.66404305136083987787137306906, 6.61140567056416146260344062097, 7.26018050726321875202504615325, 8.281602087764463646461846798646, 8.754811203143028350397849862679
