L(s) = 1 | + 2.59i·3-s + 2.72i·7-s − 3.72·9-s − 11-s + i·13-s + 4.59i·17-s + 8.18·19-s − 7.05·21-s − 0.407i·23-s − 1.87i·27-s − 7.46·29-s + 4.44·31-s − 2.59i·33-s + 7.31i·37-s − 2.59·39-s + ⋯ |
L(s) = 1 | + 1.49i·3-s + 1.02i·7-s − 1.24·9-s − 0.301·11-s + 0.277i·13-s + 1.11i·17-s + 1.87·19-s − 1.53·21-s − 0.0849i·23-s − 0.360i·27-s − 1.38·29-s + 0.798·31-s − 0.451i·33-s + 1.20i·37-s − 0.415·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.479161485\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.479161485\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 - 2.59iT - 3T^{2} \) |
| 7 | \( 1 - 2.72iT - 7T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 - 4.59iT - 17T^{2} \) |
| 19 | \( 1 - 8.18T + 19T^{2} \) |
| 23 | \( 1 + 0.407iT - 23T^{2} \) |
| 29 | \( 1 + 7.46T + 29T^{2} \) |
| 31 | \( 1 - 4.44T + 31T^{2} \) |
| 37 | \( 1 - 7.31iT - 37T^{2} \) |
| 41 | \( 1 + 3.31T + 41T^{2} \) |
| 43 | \( 1 - 7.49iT - 43T^{2} \) |
| 47 | \( 1 - 7.05iT - 47T^{2} \) |
| 53 | \( 1 - 0.979iT - 53T^{2} \) |
| 59 | \( 1 - 7.05T + 59T^{2} \) |
| 61 | \( 1 + 4.46T + 61T^{2} \) |
| 67 | \( 1 + 2.25iT - 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 12.1iT - 73T^{2} \) |
| 79 | \( 1 - 3.14T + 79T^{2} \) |
| 83 | \( 1 + 16.6iT - 83T^{2} \) |
| 89 | \( 1 + 8.27T + 89T^{2} \) |
| 97 | \( 1 + 3.03iT - 97T^{2} \) |
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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
−9.045842691401311919884005970382, −8.202948389664735272813633337050, −7.51131408673134920048320630340, −6.29942031779525338091355862853, −5.67024733313508221127761976165, −5.03251964596899350523477117121, −4.35893561377315409355128125114, −3.40504395423764516851178667167, −2.85878822126954387550321228036, −1.56866698180086515302174629208,
0.44713889193513969813333676381, 1.16525307819811943786882023230, 2.23725586533180640272723912658, 3.14116854626314402209580024666, 4.03533169277165593586457842204, 5.27513644926694229127465309369, 5.69205292261557915208089686979, 7.01603375953523008820325445034, 7.07912197132081707310228645173, 7.68423136024112236012473274769