| L(s) = 1 | − i·3-s + 2i·7-s + 2·9-s − 11-s − 4i·13-s − 2i·17-s + 2·21-s − i·23-s − 5i·27-s − 7·31-s + i·33-s + 3i·37-s − 4·39-s − 8·41-s − 6i·43-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.755i·7-s + 0.666·9-s − 0.301·11-s − 1.10i·13-s − 0.485i·17-s + 0.436·21-s − 0.208i·23-s − 0.962i·27-s − 1.25·31-s + 0.174i·33-s + 0.493i·37-s − 0.640·39-s − 1.24·41-s − 0.914i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.419019141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.419019141\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 - 3iT - 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 5T + 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 - 7iT - 67T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 + 7iT - 97T^{2} \) |
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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
−8.259006030621535316934884903817, −7.14782282991405075375376931907, −7.00602941825289942251902585716, −5.72563659315937116859868993227, −5.43151271222498705066696534751, −4.40237965223868227791761932714, −3.38291483460000137498878470416, −2.51655034540659664564190265948, −1.65904108881844393081677704161, −0.40576330652328546375250683797,
1.23830926299751393356431473401, 2.18608893983310120829240641878, 3.54728046189461839761545605800, 4.01409720753106071396164710116, 4.75390618813271717153623928527, 5.51886163905214676100774541635, 6.59101159067835030906321698276, 7.08844497532134828477364697417, 7.81322293939355045824314336652, 8.665698093272711287546339429481
