L(s) = 1 | + 3i·3-s − 12.8·5-s + (−6.32 + 17.4i)7-s − 9·9-s − 25.0·11-s + 64.1·13-s − 38.4i·15-s + 77.4i·17-s + 49.9i·19-s + (−52.2 − 18.9i)21-s + 80.5i·23-s + 39.4·25-s − 27i·27-s + 159. i·29-s − 96.5·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 1.14·5-s + (−0.341 + 0.939i)7-s − 0.333·9-s − 0.686·11-s + 1.36·13-s − 0.662i·15-s + 1.10i·17-s + 0.603i·19-s + (−0.542 − 0.197i)21-s + 0.730i·23-s + 0.315·25-s − 0.192i·27-s + 1.01i·29-s − 0.559·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.573 + 0.819i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.6079198789\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6079198789\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (6.32 - 17.4i)T \) |
good | 5 | \( 1 + 12.8T + 125T^{2} \) |
| 11 | \( 1 + 25.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 64.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 77.4iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 49.9iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 80.5iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 159. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 96.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 274. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 299. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 385.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 418.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 665. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 445. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 599.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 675.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 877. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 696. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.24e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 238. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 743. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.55e3iT - 9.12e5T^{2} \) |
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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
−9.757138939233557364413197367204, −8.773118727757241061451329307839, −8.310562583362976621523940232139, −7.56182936232505502393644615843, −6.26698163486301040574582634006, −5.68173074071985907577351719139, −4.60007834473215020187677260412, −3.63632777637254086160464050399, −3.08800688440542811384518475357, −1.52964580881445982186614903161,
0.19251003687557528203801272174, 0.814040568764020112899134334853, 2.43381136359881596265094381135, 3.56252393914999107457760957147, 4.20365993960764201164816772317, 5.35946296582097279657708531495, 6.46048255547380110812546679208, 7.21500722350856912381642820072, 7.77854985779622623314133806035, 8.512591864636519477469357743534