| L(s) = 1 | + 3.14i·3-s + 12.2·7-s − 0.877·9-s + 0.636·11-s − 19.9i·13-s − 20.7·17-s + (−4.37 + 18.4i)19-s + 38.4i·21-s + 6.16·23-s + 25.5i·27-s − 11.9i·29-s + 47.7i·31-s + 1.99i·33-s + 19.1i·37-s + 62.5·39-s + ⋯ |
| L(s) = 1 | + 1.04i·3-s + 1.74·7-s − 0.0975·9-s + 0.0578·11-s − 1.53i·13-s − 1.21·17-s + (−0.230 + 0.973i)19-s + 1.82i·21-s + 0.267·23-s + 0.945i·27-s − 0.413i·29-s + 1.54i·31-s + 0.0605i·33-s + 0.518i·37-s + 1.60·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 - 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.590566965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.590566965\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (4.37 - 18.4i)T \) |
| good | 3 | \( 1 - 3.14iT - 9T^{2} \) |
| 7 | \( 1 - 12.2T + 49T^{2} \) |
| 11 | \( 1 - 0.636T + 121T^{2} \) |
| 13 | \( 1 + 19.9iT - 169T^{2} \) |
| 17 | \( 1 + 20.7T + 289T^{2} \) |
| 23 | \( 1 - 6.16T + 529T^{2} \) |
| 29 | \( 1 + 11.9iT - 841T^{2} \) |
| 31 | \( 1 - 47.7iT - 961T^{2} \) |
| 37 | \( 1 - 19.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 22.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 34.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 75.4T + 2.20e3T^{2} \) |
| 53 | \( 1 + 87.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 40.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 10.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 16.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 23.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 103.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 123. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 82.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + 34.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 66.0iT - 9.40e3T^{2} \) |
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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
−9.155940573699478669365764046341, −8.382057057005347393766352468359, −7.905693322174481585910165859024, −6.91057563009892628709620521089, −5.65931775629317675431918688366, −5.02523771155842106699431460519, −4.39937107389156874763368678226, −3.53480217299140771065436699517, −2.27607428361574344561193184004, −1.09644981465015144533127342795,
0.76935529149324183924221269638, 1.92308004658654378642509880290, 2.27695706371269648940148779442, 4.27372938415045809702499680845, 4.54333258894510874048980733320, 5.76652060634261075386843016253, 6.72077155673499752178498809825, 7.29353320315355948167863227115, 7.919183917488954494434588063200, 8.877869146504541092189461460460
