| 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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.729338264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.729338264\) |
| \(L(2)\) |
|
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
−8.283333907757947669873537215865, −8.030568624643544302036979779258, −7.20365477278624377338969301776, −6.36157538038380662959137314341, −5.67235243909531580680856408434, −4.60031866690565513880670579804, −3.69105918730999876530182658645, −2.27699923536294836578271579918, −1.61549287917761063573478997258, −0.43650133033859319990376703056,
1.41790005864460272246218372347, 2.49701859030550024407408852040, 3.99646000389235232746774059255, 4.25118374382459368148639957114, 5.12264613602001905615886436185, 6.09214333107184675418252919440, 7.00198138161208505443960999278, 7.926260921809397181645631831952, 8.777759484540039237905362525429, 9.322758943280324637122341789086
