| L(s) = 1 | − i·3-s + (−2.23 − 0.123i)5-s + 4.96i·7-s − 9-s + 3.21·11-s + i·13-s + (−0.123 + 2.23i)15-s + 0.448i·17-s − 7.23·19-s + 4.96·21-s + 6.26i·23-s + (4.96 + 0.551i)25-s + i·27-s + 2.21·29-s − 2.19·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (−0.998 − 0.0552i)5-s + 1.87i·7-s − 0.333·9-s + 0.969·11-s + 0.277i·13-s + (−0.0318 + 0.576i)15-s + 0.108i·17-s − 1.65·19-s + 1.08·21-s + 1.30i·23-s + (0.993 + 0.110i)25-s + 0.192i·27-s + 0.411·29-s − 0.393·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0552i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0552i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2991808279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2991808279\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (2.23 + 0.123i)T \) |
| 13 | \( 1 - iT \) |
| good | 7 | \( 1 - 4.96iT - 7T^{2} \) |
| 11 | \( 1 - 3.21T + 11T^{2} \) |
| 17 | \( 1 - 0.448iT - 17T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 23 | \( 1 - 6.26iT - 23T^{2} \) |
| 29 | \( 1 - 2.21T + 29T^{2} \) |
| 31 | \( 1 + 2.19T + 31T^{2} \) |
| 37 | \( 1 + 8.26iT - 37T^{2} \) |
| 41 | \( 1 - 1.79T + 41T^{2} \) |
| 43 | \( 1 + 6.21iT - 43T^{2} \) |
| 47 | \( 1 - 1.66iT - 47T^{2} \) |
| 53 | \( 1 - 1.16iT - 53T^{2} \) |
| 59 | \( 1 + 5.88T + 59T^{2} \) |
| 61 | \( 1 + 1.76T + 61T^{2} \) |
| 67 | \( 1 + 2.73iT - 67T^{2} \) |
| 71 | \( 1 + 4.11T + 71T^{2} \) |
| 73 | \( 1 + 10.4iT - 73T^{2} \) |
| 79 | \( 1 - 1.05T + 79T^{2} \) |
| 83 | \( 1 - 5.77iT - 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 - 8.37iT - 97T^{2} \) |
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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
−8.986284418959545494586171071038, −8.382812965423893388235185044470, −7.64414816924517949297825628880, −6.75865544281144250171121239764, −6.10363703095570340301299132520, −5.38226593709758955899364811118, −4.34507083097570378413009605936, −3.49753326608845628194564292583, −2.45509130450739897236862393247, −1.61667179562945360622973752273,
0.10026819716979011853267926234, 1.20569078545980131390234970301, 2.89055005213176785361286015385, 3.85779466151119133041911420480, 4.26945261970642875935702044399, 4.80133285094503026496808511402, 6.39923908015830491740273764979, 6.73151231933284239195336854602, 7.64005931844114677156456782665, 8.300721928268269717445174571926
