| L(s) = 1 | − 4.38·5-s + i·7-s − 5.11i·11-s − 5.21i·13-s − 2i·17-s + 6.50·19-s − 4.63·23-s + 14.2·25-s − 1.26·29-s − 4.21i·31-s − 4.38i·35-s − 1.98i·37-s + 7.37i·41-s − 3.71·43-s + 2.57·47-s + ⋯ |
| L(s) = 1 | − 1.96·5-s + 0.377i·7-s − 1.54i·11-s − 1.44i·13-s − 0.485i·17-s + 1.49·19-s − 0.967·23-s + 2.84·25-s − 0.235·29-s − 0.756i·31-s − 0.741i·35-s − 0.326i·37-s + 1.15i·41-s − 0.566·43-s + 0.375·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6898836696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6898836696\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 5 | \( 1 + 4.38T + 5T^{2} \) |
| 11 | \( 1 + 5.11iT - 11T^{2} \) |
| 13 | \( 1 + 5.21iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 6.50T + 19T^{2} \) |
| 23 | \( 1 + 4.63T + 23T^{2} \) |
| 29 | \( 1 + 1.26T + 29T^{2} \) |
| 31 | \( 1 + 4.21iT - 31T^{2} \) |
| 37 | \( 1 + 1.98iT - 37T^{2} \) |
| 41 | \( 1 - 7.37iT - 41T^{2} \) |
| 43 | \( 1 + 3.71T + 43T^{2} \) |
| 47 | \( 1 - 2.57T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 - 3.50iT - 59T^{2} \) |
| 61 | \( 1 + 4.53iT - 61T^{2} \) |
| 67 | \( 1 + 5.21T + 67T^{2} \) |
| 71 | \( 1 + 1.36T + 71T^{2} \) |
| 73 | \( 1 - 15.4T + 73T^{2} \) |
| 79 | \( 1 + 3.98iT - 79T^{2} \) |
| 83 | \( 1 + 11.6iT - 83T^{2} \) |
| 89 | \( 1 + 14.8iT - 89T^{2} \) |
| 97 | \( 1 - 4.42T + 97T^{2} \) |
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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
−7.68289081263011908368473037887, −7.46009429839029910832090577450, −6.27331464782272546098620327076, −5.55657607911539603324883638313, −4.88011371309827464802949361751, −3.84534072736078881482749881481, −3.32141790107222752529798829616, −2.77813460786439712269812502144, −0.913878713247935390473616394579, −0.25226567411892450093923872965,
1.17025935919739824696102865919, 2.27117824727112993379756188076, 3.54360242059474572640000955966, 3.98554327727296136659691997215, 4.57427676928286667859099335084, 5.29881689386248716702168085242, 6.71084079318939738881705290084, 7.06348829800172119126170002747, 7.60376856668238904078123435178, 8.222106692137613185890237380902
