| L(s) = 1 | + 3-s − 4i·5-s + 4i·7-s + 9-s + 2i·11-s − 4i·15-s + 6·17-s + 4i·19-s + 4i·21-s − 4·23-s − 11·25-s + 27-s − 6·29-s + 8i·31-s + 2i·33-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78i·5-s + 1.51i·7-s + 0.333·9-s + 0.603i·11-s − 1.03i·15-s + 1.45·17-s + 0.917i·19-s + 0.872i·21-s − 0.834·23-s − 2.20·25-s + 0.192·27-s − 1.11·29-s + 1.43i·31-s + 0.348i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.028662942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.028662942\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 4iT - 5T^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 8iT - 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 + 4iT - 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 10iT - 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 10iT - 83T^{2} \) |
| 89 | \( 1 + 8iT - 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.598521452864360955829862839767, −8.062544245983616492325864985018, −7.39745331628988825392471511402, −6.00349448940733312872871203508, −5.54753876115079261667719401602, −4.88763169369792174593110492005, −4.04329914482921857822801261330, −3.06956472516852432533359385506, −1.92788264923453754657444941782, −1.30503241743468739708169986950,
0.53877028193135650714978210832, 2.02473872664100069897206355721, 2.94033016352601970521626431241, 3.70695839801149713489726659863, 4.04211585991692898604170277347, 5.50894190077442623009113207277, 6.30424164455667052840375526384, 7.02595923762181946990791318183, 7.71717408960355757698576197183, 7.80135093654205926373464353434
