L(s) = 1 | − 3·3-s − 4.47i·5-s + (14.9 − 10.8i)7-s + 9·9-s + 7.61i·11-s − 13.3i·13-s + 13.4i·15-s + 55.3i·17-s + 73.9·19-s + (−44.9 + 32.6i)21-s + 133. i·23-s + 104.·25-s − 27·27-s − 23.3·29-s − 241.·31-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.400i·5-s + (0.808 − 0.587i)7-s + 0.333·9-s + 0.208i·11-s − 0.285i·13-s + 0.231i·15-s + 0.789i·17-s + 0.892·19-s + (−0.467 + 0.339i)21-s + 1.20i·23-s + 0.839·25-s − 0.192·27-s − 0.149·29-s − 1.39·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 - 0.587i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.808 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.719334107\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.719334107\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 + (-14.9 + 10.8i)T \) |
good | 5 | \( 1 + 4.47iT - 125T^{2} \) |
| 11 | \( 1 - 7.61iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 13.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 55.3iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 73.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 133. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 23.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 241.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 178.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 494. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 72.6iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 59.7T + 1.03e5T^{2} \) |
| 53 | \( 1 - 569.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 59.5T + 2.05e5T^{2} \) |
| 61 | \( 1 - 629. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 599. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 407. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 680. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.08e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 935.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 12.1iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.43e3iT - 9.12e5T^{2} \) |
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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
−9.349989225691462026937067511256, −8.475333467577384681864945531606, −7.58230368084627401548569711732, −7.03726935347128281208415276472, −5.79154892186140663566839571157, −5.18453642296795208179271202501, −4.32727364344263285162119361841, −3.36313047743202579362540308292, −1.73653638320281169260902803495, −0.953987175299211555789793835914,
0.52145347446160714668521918472, 1.83852687413677039519228246482, 2.88870782909482381212992105503, 4.11008824898193398610747184788, 5.16610292831680297265339537127, 5.62268589008880664874581299322, 6.83124675344713750468765444753, 7.32078879568033157782400703624, 8.454022438034771464726549106652, 9.087846684215149212967884759698