| L(s) = 1 | − 4.83i·3-s + 3.72·7-s − 14.3·9-s − 13.8·11-s + 4.10i·13-s + 15.3·17-s + (−16.4 + 9.49i)19-s − 18.0i·21-s − 17.4·23-s + 25.8i·27-s + 16.3i·29-s + 15.9i·31-s + 67.1i·33-s + 5.14i·37-s + 19.8·39-s + ⋯ |
| L(s) = 1 | − 1.61i·3-s + 0.532·7-s − 1.59·9-s − 1.26·11-s + 0.315i·13-s + 0.900·17-s + (−0.866 + 0.499i)19-s − 0.857i·21-s − 0.758·23-s + 0.956i·27-s + 0.565i·29-s + 0.513i·31-s + 2.03i·33-s + 0.139i·37-s + 0.508·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9609528886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9609528886\) |
| \(L(2)\) |
|
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 | \( 1 + (16.4 - 9.49i)T \) |
| good | 3 | \( 1 + 4.83iT - 9T^{2} \) |
| 7 | \( 1 - 3.72T + 49T^{2} \) |
| 11 | \( 1 + 13.8T + 121T^{2} \) |
| 13 | \( 1 - 4.10iT - 169T^{2} \) |
| 17 | \( 1 - 15.3T + 289T^{2} \) |
| 23 | \( 1 + 17.4T + 529T^{2} \) |
| 29 | \( 1 - 16.3iT - 841T^{2} \) |
| 31 | \( 1 - 15.9iT - 961T^{2} \) |
| 37 | \( 1 - 5.14iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 42.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.56T + 1.84e3T^{2} \) |
| 47 | \( 1 - 54.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 55.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 101. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 80.5T + 3.72e3T^{2} \) |
| 67 | \( 1 - 24.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 121. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 33.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 102. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 80.3T + 6.88e3T^{2} \) |
| 89 | \( 1 + 12.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 161. iT - 9.40e3T^{2} \) |
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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
−8.635775661269814344567969209932, −8.230907374841830304313990944774, −7.47132617381893156868473671891, −6.96804677663551513341877645225, −5.89232702427394258060849607842, −5.39885366246647893775953476746, −4.13638845570424447184480773414, −2.79352228922739474517844560352, −2.01206619197662364495041976927, −1.06807360770939907704786158046,
0.26277180490265699938069111109, 2.20050112133075168455679271393, 3.20465474674124923117577834754, 4.08816056012925510587869868185, 4.91498199528241147841051900626, 5.40767752076327899923607703321, 6.35393332661997360611347391135, 7.80313416217702916540117616481, 8.148109031950071442450013219343, 9.119026351398966002427319131347
