L(s) = 1 | − 3i·3-s + 20.7·5-s + (11.8 − 14.2i)7-s − 9·9-s + 39.2·11-s − 36.2·13-s − 62.1i·15-s + 92.2i·17-s − 87.5i·19-s + (−42.7 − 35.4i)21-s + 100. i·23-s + 303.·25-s + 27i·27-s + 36.2i·29-s + 264.·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 1.85·5-s + (0.638 − 0.770i)7-s − 0.333·9-s + 1.07·11-s − 0.773·13-s − 1.06i·15-s + 1.31i·17-s − 1.05i·19-s + (−0.444 − 0.368i)21-s + 0.915i·23-s + 2.42·25-s + 0.192i·27-s + 0.231i·29-s + 1.53·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.578 + 0.815i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.769084860\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.769084860\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| 7 | \( 1 + (-11.8 + 14.2i)T \) |
good | 5 | \( 1 - 20.7T + 125T^{2} \) |
| 11 | \( 1 - 39.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 36.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 92.2iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 87.5iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 100. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 36.2iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 264.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 229. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 337. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 368.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 314.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 108. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 185. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 163.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 458.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 584. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 1.09e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 438. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 722. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 452. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 597. iT - 9.12e5T^{2} \) |
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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
−9.185377761910238024772721636970, −8.408208870525879789357346070195, −7.26533939615623562773187264773, −6.67005756118835134289328784799, −5.88133396522625747467145640780, −5.08076331722393805189276761769, −4.00840181036604916734686352236, −2.55149077981229095361476917067, −1.71209902423568012099946589579, −0.965337125363911928638868159859,
1.17374631734500170152142682964, 2.22012586847288549301858914558, 2.95526028885130660285498385980, 4.62324127081206352758985129872, 5.06302786870521896846341043051, 6.11122769170163184451055764703, 6.50503086350112381577585164750, 7.912396490202562387953814224423, 8.854710651429702548444672455506, 9.518093331290070342515016731252