| L(s) = 1 | + (−1.99 − 1.00i)5-s + (−2.58 − 0.542i)7-s + 1.87i·11-s − 3.32·13-s − 0.849i·17-s − 1.25i·19-s − 1.04·23-s + (2.96 + 4.02i)25-s + 6.04i·29-s − 4.77i·31-s + (4.62 + 3.69i)35-s − 1.93i·37-s + 0.107·41-s − 6.07i·43-s + 6.96i·47-s + ⋯ |
| L(s) = 1 | + (−0.892 − 0.450i)5-s + (−0.978 − 0.205i)7-s + 0.564i·11-s − 0.922·13-s − 0.206i·17-s − 0.287i·19-s − 0.218·23-s + (0.593 + 0.804i)25-s + 1.12i·29-s − 0.856i·31-s + (0.781 + 0.624i)35-s − 0.317i·37-s + 0.0168·41-s − 0.925i·43-s + 1.01i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9017859144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9017859144\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.99 + 1.00i)T \) |
| 7 | \( 1 + (2.58 + 0.542i)T \) |
| good | 11 | \( 1 - 1.87iT - 11T^{2} \) |
| 13 | \( 1 + 3.32T + 13T^{2} \) |
| 17 | \( 1 + 0.849iT - 17T^{2} \) |
| 19 | \( 1 + 1.25iT - 19T^{2} \) |
| 23 | \( 1 + 1.04T + 23T^{2} \) |
| 29 | \( 1 - 6.04iT - 29T^{2} \) |
| 31 | \( 1 + 4.77iT - 31T^{2} \) |
| 37 | \( 1 + 1.93iT - 37T^{2} \) |
| 41 | \( 1 - 0.107T + 41T^{2} \) |
| 43 | \( 1 + 6.07iT - 43T^{2} \) |
| 47 | \( 1 - 6.96iT - 47T^{2} \) |
| 53 | \( 1 - 5.77T + 53T^{2} \) |
| 59 | \( 1 - 8.15T + 59T^{2} \) |
| 61 | \( 1 - 1.84iT - 61T^{2} \) |
| 67 | \( 1 + 7.06iT - 67T^{2} \) |
| 71 | \( 1 - 16.1iT - 71T^{2} \) |
| 73 | \( 1 - 7.51T + 73T^{2} \) |
| 79 | \( 1 + 1.19T + 79T^{2} \) |
| 83 | \( 1 + 2.48iT - 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 1.21T + 97T^{2} \) |
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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
−9.024191278502593397164998272515, −8.150335434623622675858506979559, −7.22228442164724230322931426224, −6.99370805367161791325225601232, −5.78568821835853592022507041155, −4.90093749052335201560034816896, −4.13821994413241614857282024553, −3.31713977093632139210793174068, −2.28286208193320221006284020755, −0.67995693371257145395104364783,
0.48887611698405795022316850826, 2.32296888881150131639191974780, 3.20701188160475529855651666069, 3.88197905980832393117374579962, 4.88025851994463467751077804267, 5.91481407983955881019196268246, 6.62190852553575713040414362253, 7.32397347701246782118455844392, 8.100577257799645905217303768257, 8.784149949487374949072058401952
