L(s) = 1 | + 3i·3-s − 15.6·5-s + (−10.8 + 15.0i)7-s − 9·9-s + 41.3·11-s − 8.02·13-s − 47.0i·15-s + 14.2i·17-s − 10.2i·19-s + (−45.0 − 32.5i)21-s − 203. i·23-s + 121.·25-s − 27i·27-s − 82.6i·29-s + 156.·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 1.40·5-s + (−0.585 + 0.810i)7-s − 0.333·9-s + 1.13·11-s − 0.171·13-s − 0.810i·15-s + 0.202i·17-s − 0.123i·19-s + (−0.468 − 0.337i)21-s − 1.84i·23-s + 0.969·25-s − 0.192i·27-s − 0.529i·29-s + 0.905·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.355 - 0.934i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.355 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.157607799\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.157607799\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (10.8 - 15.0i)T \) |
good | 5 | \( 1 + 15.6T + 125T^{2} \) |
| 11 | \( 1 - 41.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 8.02T + 2.19e3T^{2} \) |
| 17 | \( 1 - 14.2iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 10.2iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 203. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 82.6iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 156.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 106. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 315. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 154.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 474.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 266. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 425. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 8.30T + 2.26e5T^{2} \) |
| 67 | \( 1 - 820.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 109. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 53.8iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.17e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 507. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.06e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 573. iT - 9.12e5T^{2} \) |
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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
−9.396906051153358205789917720490, −8.487922759336698254961530840633, −8.087463313586238786256461597916, −6.76838910941503057042527771014, −6.27937776109648199341620854397, −4.96844233683508272087355374123, −4.18596802307453414257640519605, −3.46678869024292099437041135142, −2.45319352698250395151211133912, −0.63896254204605106055788561395,
0.47315616275243775188083278770, 1.47394348748613471969318116349, 3.18357127920071955968996670156, 3.76445986648910293752910474100, 4.63868312488657841298738636871, 5.95324291738497182225894414766, 6.92528699061425457591362039757, 7.35353858521797006307491866877, 8.085822262895768401304840072253, 9.020101418035666833857887993827