| L(s) = 1 | − 1.73i·3-s + (1.77 + 1.35i)5-s + 3.32i·7-s − 0.0262·9-s − 5.77·11-s + 1.10i·13-s + (2.36 − 3.09i)15-s − 0.893i·17-s + 2.42·19-s + 5.77·21-s − i·23-s + (1.31 + 4.82i)25-s − 5.17i·27-s − 4.11·29-s + 9.54·31-s + ⋯ |
| L(s) = 1 | − 1.00i·3-s + (0.794 + 0.606i)5-s + 1.25i·7-s − 0.00874·9-s − 1.74·11-s + 0.305i·13-s + (0.609 − 0.798i)15-s − 0.216i·17-s + 0.557·19-s + 1.26·21-s − 0.208i·23-s + (0.263 + 0.964i)25-s − 0.995i·27-s − 0.763·29-s + 1.71·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.606 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.663703352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.663703352\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.77 - 1.35i)T \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 + 1.73iT - 3T^{2} \) |
| 7 | \( 1 - 3.32iT - 7T^{2} \) |
| 11 | \( 1 + 5.77T + 11T^{2} \) |
| 13 | \( 1 - 1.10iT - 13T^{2} \) |
| 17 | \( 1 + 0.893iT - 17T^{2} \) |
| 19 | \( 1 - 2.42T + 19T^{2} \) |
| 29 | \( 1 + 4.11T + 29T^{2} \) |
| 31 | \( 1 - 9.54T + 31T^{2} \) |
| 37 | \( 1 - 7.69iT - 37T^{2} \) |
| 41 | \( 1 - 0.00418T + 41T^{2} \) |
| 43 | \( 1 - 9.97iT - 43T^{2} \) |
| 47 | \( 1 - 10.0iT - 47T^{2} \) |
| 53 | \( 1 - 6.25iT - 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 - 10.9iT - 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 - 1.89iT - 73T^{2} \) |
| 79 | \( 1 + 0.216T + 79T^{2} \) |
| 83 | \( 1 - 5.38iT - 83T^{2} \) |
| 89 | \( 1 + 6.00T + 89T^{2} \) |
| 97 | \( 1 + 2.08iT - 97T^{2} \) |
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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
−9.515628176570869802071674575506, −8.343582110378367154151212665189, −7.82259911577162600428616803572, −6.94154221261087124464335045623, −6.18577910550738521288206266715, −5.56375370519522376794678580970, −4.69786240238372450098900259202, −2.75791025198023463508048543064, −2.61969864924976145340974369367, −1.41072135574155377755105233294,
0.61633565659423295991926050197, 2.11450080795921441732554050191, 3.38339236175052042735170962813, 4.24453503414736375122268429729, 5.12191010301714677020391948679, 5.51199741000344436789593700853, 6.80997854947556887014755154929, 7.66600764078904222135991424803, 8.372218697889308812819392499498, 9.413878274918042554301140653044
