L(s) = 1 | + 2.63i·5-s + 7-s − 4.29i·11-s − 0.0480i·13-s + 3.16·17-s + 0.726i·19-s − 5.28·23-s − 1.94·25-s + 8.00i·29-s − 4.90·31-s + 2.63i·35-s + 9.10i·37-s − 1.45·41-s − 8.85i·43-s − 9.23·47-s + ⋯ |
L(s) = 1 | + 1.17i·5-s + 0.377·7-s − 1.29i·11-s − 0.0133i·13-s + 0.766·17-s + 0.166i·19-s − 1.10·23-s − 0.388·25-s + 1.48i·29-s − 0.881·31-s + 0.445i·35-s + 1.49i·37-s − 0.227·41-s − 1.35i·43-s − 1.34·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 - 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.320386908\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.320386908\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 5 | \( 1 - 2.63iT - 5T^{2} \) |
| 11 | \( 1 + 4.29iT - 11T^{2} \) |
| 13 | \( 1 + 0.0480iT - 13T^{2} \) |
| 17 | \( 1 - 3.16T + 17T^{2} \) |
| 19 | \( 1 - 0.726iT - 19T^{2} \) |
| 23 | \( 1 + 5.28T + 23T^{2} \) |
| 29 | \( 1 - 8.00iT - 29T^{2} \) |
| 31 | \( 1 + 4.90T + 31T^{2} \) |
| 37 | \( 1 - 9.10iT - 37T^{2} \) |
| 41 | \( 1 + 1.45T + 41T^{2} \) |
| 43 | \( 1 + 8.85iT - 43T^{2} \) |
| 47 | \( 1 + 9.23T + 47T^{2} \) |
| 53 | \( 1 - 3.14iT - 53T^{2} \) |
| 59 | \( 1 - 5.90iT - 59T^{2} \) |
| 61 | \( 1 + 2.19iT - 61T^{2} \) |
| 67 | \( 1 - 8.55iT - 67T^{2} \) |
| 71 | \( 1 + 3.86T + 71T^{2} \) |
| 73 | \( 1 + 7.56T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 - 9.05iT - 83T^{2} \) |
| 89 | \( 1 + 5.15T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−8.328893380298959425172523047336, −7.55239310529290695861714318713, −6.94228195472977108140096190366, −6.17492527449638971786375457731, −5.61861644267383883158577897390, −4.78226154794527299546290543008, −3.48688174518002179011681708830, −3.36276281660016150273733665341, −2.27106449245938900064045305917, −1.19383714320830173237702930981,
0.34072835190528893378823779008, 1.57820048152324375141538590388, 2.17279197961991589968550523824, 3.50205666043321454314829952789, 4.39898629123723135991886196665, 4.80352071398497189227832450305, 5.59961096177678184546818168865, 6.30166945846588969112820665004, 7.35853533140055063477488365705, 7.83579883963515812775714554199