L(s) = 1 | − 3i·3-s + 13.9i·5-s − 7·7-s − 9·9-s + 44.5i·11-s + 81.2i·13-s + 41.7·15-s − 136.·17-s + 114. i·19-s + 21i·21-s − 67.0·23-s − 69.1·25-s + 27i·27-s − 222. i·29-s − 135.·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 1.24i·5-s − 0.377·7-s − 0.333·9-s + 1.22i·11-s + 1.73i·13-s + 0.719·15-s − 1.95·17-s + 1.38i·19-s + 0.218i·21-s − 0.607·23-s − 0.552·25-s + 0.192i·27-s − 1.42i·29-s − 0.783·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.3615036572\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3615036572\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + 7T \) |
good | 5 | \( 1 - 13.9iT - 125T^{2} \) |
| 11 | \( 1 - 44.5iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 81.2iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 136.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 114. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 67.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 222. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 135.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 298. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 88.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 241. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 377.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 174. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 26.8iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 51.2iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 1.01e3iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 502.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 709.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 716.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 882. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.29e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 386.T + 9.12e5T^{2} \) |
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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
−9.643699507835044936970606436246, −9.112553233674279434637196691547, −7.88450582184420952271760639904, −7.12879973437637192139031535011, −6.59769580251564496532044497324, −6.00905868015935574586518961818, −4.44439711362793074792258801854, −3.79290876898548648575534203417, −2.23831170057691249750381976328, −2.02279439248967130940336303895,
0.096988640571687690146555158920, 0.882281362236881391751586306063, 2.57683719925728563305728870114, 3.51642728786636743106350654466, 4.58428034054697365493653346135, 5.26967451657238582252443795651, 6.01179132698168806809881782004, 7.10102889581202305781669279960, 8.318783186290126939169237424495, 8.793948491340410821540282360160