| L(s) = 1 | + 1.78·3-s + (0.615 − 2.14i)5-s + 2.64·7-s + 0.171·9-s − 0.737i·13-s + (1.09 − 3.82i)15-s − 5.43i·19-s + 4.71·21-s + 7.48·23-s + (−4.24 − 2.64i)25-s − 5.03·27-s + (1.62 − 5.68i)35-s − 1.31i·39-s + (0.105 − 0.368i)45-s + 7.00·49-s + ⋯ |
| L(s) = 1 | + 1.02·3-s + (0.275 − 0.961i)5-s + 0.999·7-s + 0.0571·9-s − 0.204i·13-s + (0.282 − 0.988i)15-s − 1.24i·19-s + 1.02·21-s + 1.56·23-s + (−0.848 − 0.529i)25-s − 0.969·27-s + (0.275 − 0.961i)35-s − 0.210i·39-s + (0.0157 − 0.0549i)45-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.921134688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.921134688\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.615 + 2.14i)T \) |
| 7 | \( 1 - 2.64T \) |
| good | 3 | \( 1 - 1.78T + 3T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 0.737iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 5.43iT - 19T^{2} \) |
| 23 | \( 1 - 7.48T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 6.45iT - 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 5.65iT - 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 16.9iT - 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−8.928352094338480650464058192571, −8.271322654398524025085938727876, −7.64512353731259801144647283934, −6.72054283917570052024644046082, −5.44329138475534121904568982783, −4.95878289586431557122246212830, −4.05159180473125003111078593482, −2.91403198814185426364770428473, −2.06242476762599219700006384843, −0.924653059221757662986693068396,
1.54453802420477711684025711678, 2.44636889364079633629853601227, 3.25105128203615283604675722166, 4.09851338727506172833228741676, 5.23741864916780912342272715911, 6.04291334943806461869340744968, 7.09679984613279048575277695573, 7.65051910015022453591281008249, 8.423619890687896824039653950777, 9.011911519197927330065893141029
