| L(s) = 1 | − 3.26i·3-s − 6.30·7-s − 1.67·9-s + 4.53·11-s − 11.5i·13-s + 1.08·17-s + (−16.8 − 8.76i)19-s + 20.5i·21-s + 31.5·23-s − 23.9i·27-s + 38.1i·29-s − 58.2i·31-s − 14.8i·33-s − 40.2i·37-s − 37.6·39-s + ⋯ |
| L(s) = 1 | − 1.08i·3-s − 0.900·7-s − 0.186·9-s + 0.411·11-s − 0.887i·13-s + 0.0638·17-s + (−0.887 − 0.461i)19-s + 0.980i·21-s + 1.37·23-s − 0.886i·27-s + 1.31i·29-s − 1.87i·31-s − 0.448i·33-s − 1.08i·37-s − 0.966·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 - 0.461i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.887 - 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7943405928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7943405928\) |
| \(L(2)\) |
|
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 | \( 1 + (16.8 + 8.76i)T \) |
| good | 3 | \( 1 + 3.26iT - 9T^{2} \) |
| 7 | \( 1 + 6.30T + 49T^{2} \) |
| 11 | \( 1 - 4.53T + 121T^{2} \) |
| 13 | \( 1 + 11.5iT - 169T^{2} \) |
| 17 | \( 1 - 1.08T + 289T^{2} \) |
| 23 | \( 1 - 31.5T + 529T^{2} \) |
| 29 | \( 1 - 38.1iT - 841T^{2} \) |
| 31 | \( 1 + 58.2iT - 961T^{2} \) |
| 37 | \( 1 + 40.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 18.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 2.74T + 1.84e3T^{2} \) |
| 47 | \( 1 + 76.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 8.06iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 95.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 4.28T + 3.72e3T^{2} \) |
| 67 | \( 1 + 70.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 93.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 14.2T + 5.32e3T^{2} \) |
| 79 | \( 1 + 6.11iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 89.0T + 6.88e3T^{2} \) |
| 89 | \( 1 + 92.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 64.6iT - 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.518002549051747262241130773149, −7.63585425232029434911224233030, −6.96678438707726093973042671108, −6.38905955929447894422716709620, −5.58777454486169122331590218181, −4.43310432335708252201017849662, −3.31970296462622237437164092086, −2.45612805456946558020349193832, −1.25055235908336167298562109090, −0.21270206689971456339766740622,
1.52248615125040244008062970666, 2.96747801637960726661331771098, 3.73640246585525173354384958239, 4.51263332915963624397151749571, 5.26326344436430926035092367289, 6.56372326360816189271467810956, 6.74713884154150983777973171450, 8.114709446093331676576783689592, 8.968715226965017112856490004061, 9.514067520586489162457341849379
