| L(s) = 1 | − 3.06i·2-s − 5.37·4-s − 5.25i·5-s − 2.64·7-s + 4.22i·8-s − 16.0·10-s − 4.48i·11-s − 6.53·13-s + 8.09i·14-s − 8.58·16-s − 18.5i·17-s − 4.12·19-s + 28.2i·20-s − 13.7·22-s − 35.9i·23-s + ⋯ |
| L(s) = 1 | − 1.53i·2-s − 1.34·4-s − 1.05i·5-s − 0.377·7-s + 0.527i·8-s − 1.60·10-s − 0.407i·11-s − 0.502·13-s + 0.578i·14-s − 0.536·16-s − 1.08i·17-s − 0.217·19-s + 1.41i·20-s − 0.623·22-s − 1.56i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6776594018\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6776594018\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 127 | \( 1 - 11.2T \) |
| good | 2 | \( 1 + 3.06iT - 4T^{2} \) |
| 5 | \( 1 + 5.25iT - 25T^{2} \) |
| 7 | \( 1 + 2.64T + 49T^{2} \) |
| 11 | \( 1 + 4.48iT - 121T^{2} \) |
| 13 | \( 1 + 6.53T + 169T^{2} \) |
| 17 | \( 1 + 18.5iT - 289T^{2} \) |
| 19 | \( 1 + 4.12T + 361T^{2} \) |
| 23 | \( 1 + 35.9iT - 529T^{2} \) |
| 29 | \( 1 - 36.9iT - 841T^{2} \) |
| 31 | \( 1 + 31.4T + 961T^{2} \) |
| 37 | \( 1 - 60.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + 19.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 19.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 61.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 46.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 24.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 43.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 6.75T + 4.48e3T^{2} \) |
| 71 | \( 1 + 44.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 8.96T + 5.32e3T^{2} \) |
| 79 | \( 1 + 23.4T + 6.24e3T^{2} \) |
| 83 | \( 1 - 60.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 43.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 64.0T + 9.40e3T^{2} \) |
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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
−9.168799242039706012845075220796, −8.572939120945185738821046725099, −7.38470450354064946361259795165, −6.27837755115642699825973260441, −4.98548138989538479637782861930, −4.46756087784567038064706283713, −3.27768168205619433918342221896, −2.43693415997748848305184857476, −1.15784122620622261227980084900, −0.21837358173725613302643886774,
2.11591015363576554829077115757, 3.42344931656166719453209156725, 4.47848269967726184743982001973, 5.61672282901082439229936191782, 6.24876810843408971934202005852, 7.00008845063717882694389393973, 7.60015396657243236792326177744, 8.328489733380142835361068204707, 9.439214332697733906987552600395, 10.00247735821383841079442301147
