| L(s) = 1 | − 2.23i·5-s + 4.47i·11-s + 6.32·13-s + 2.82i·17-s + 5.65i·23-s − 5.00·25-s − 4.47i·29-s − 2·31-s + 6.32·37-s + 12.6·43-s − 11.3i·47-s + 7·49-s + 10.0·55-s + 4.47i·59-s − 14.1i·65-s + ⋯ |
| L(s) = 1 | − 0.999i·5-s + 1.34i·11-s + 1.75·13-s + 0.685i·17-s + 1.17i·23-s − 1.00·25-s − 0.830i·29-s − 0.359·31-s + 1.03·37-s + 1.92·43-s − 1.65i·47-s + 49-s + 1.34·55-s + 0.582i·59-s − 1.75i·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.802087535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.802087535\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 4.47iT - 11T^{2} \) |
| 13 | \( 1 - 6.32T + 13T^{2} \) |
| 17 | \( 1 - 2.82iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 5.65iT - 23T^{2} \) |
| 29 | \( 1 + 4.47iT - 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 6.32T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 12.6T + 43T^{2} \) |
| 47 | \( 1 + 11.3iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 4.47iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.440146114013740180990301109301, −8.767341131608297007091476439390, −8.005955255923395951615071337488, −7.21975310780845216080540643005, −6.06856446259294401566038788815, −5.46944531943595131855348558768, −4.29472951660015688540192594650, −3.79332528982123476531233410226, −2.11746202175756783138415160715, −1.11643076745560571118496029730,
0.936493961482287285043418505234, 2.59089631277652421721084906641, 3.37624550060448332713410436372, 4.24671639553642794232156331343, 5.71937949384951853191360815508, 6.17131752747359664581299000939, 7.01191455174128541448942172635, 7.988084888825951809604603184953, 8.698522274656168743112359589275, 9.451474207782909092459335681750
