| L(s) = 1 | + 3.12·5-s − i·7-s − 0.397i·11-s − 6.55i·13-s − 2i·17-s + 0.527·19-s + 8.21·23-s + 4.79·25-s + 4.73·29-s − 7.55i·31-s − 3.12i·35-s + 3.35i·37-s + 7.48i·41-s − 1.62·43-s − 10.4·47-s + ⋯ |
| L(s) = 1 | + 1.39·5-s − 0.377i·7-s − 0.119i·11-s − 1.81i·13-s − 0.485i·17-s + 0.121·19-s + 1.71·23-s + 0.959·25-s + 0.878·29-s − 1.35i·31-s − 0.529i·35-s + 0.551i·37-s + 1.16i·41-s − 0.247·43-s − 1.52·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.665408149\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.665408149\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.12T + 5T^{2} \) |
| 11 | \( 1 + 0.397iT - 11T^{2} \) |
| 13 | \( 1 + 6.55iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 0.527T + 19T^{2} \) |
| 23 | \( 1 - 8.21T + 23T^{2} \) |
| 29 | \( 1 - 4.73T + 29T^{2} \) |
| 31 | \( 1 + 7.55iT - 31T^{2} \) |
| 37 | \( 1 - 3.35iT - 37T^{2} \) |
| 41 | \( 1 - 7.48iT - 41T^{2} \) |
| 43 | \( 1 + 1.62T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 0.795T + 53T^{2} \) |
| 59 | \( 1 + 2.47iT - 59T^{2} \) |
| 61 | \( 1 - 11.4iT - 61T^{2} \) |
| 67 | \( 1 - 6.55T + 67T^{2} \) |
| 71 | \( 1 + 2.21T + 71T^{2} \) |
| 73 | \( 1 + 5.75T + 73T^{2} \) |
| 79 | \( 1 + 10.2iT - 79T^{2} \) |
| 83 | \( 1 - 8.36iT - 83T^{2} \) |
| 89 | \( 1 + 5.31iT - 89T^{2} \) |
| 97 | \( 1 + 19.1T + 97T^{2} \) |
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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
−7.996161403845006617307037565467, −7.14132128386955349209459667060, −6.45268466320436065920339825700, −5.71929738361772347771439174516, −5.20911482429952152931841952431, −4.46547835913755809712308136802, −3.04637826612850192862577202271, −2.84635270026773653283050701686, −1.54508478891026504594513082786, −0.67620212561380455741458993931,
1.32579090498005799832329908452, 1.96520035562596015841153736045, 2.78314527451617909408218542862, 3.79191076287234359227663311870, 4.89517991328264592488040284192, 5.21783720090348525711149121324, 6.30097606688172575276148504550, 6.60893684625020305151068516995, 7.29728645915343565804770231041, 8.532487316979400724536538121637
