L(s) = 1 | − 3i·3-s − 1.64·5-s + (15.2 − 10.4i)7-s − 9·9-s + 20.3·11-s − 13.0·13-s + 4.92i·15-s + 23.9i·17-s − 87.7i·19-s + (−31.4 − 45.8i)21-s − 73.6i·23-s − 122.·25-s + 27i·27-s + 58.9i·29-s − 124.·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.146·5-s + (0.824 − 0.565i)7-s − 0.333·9-s + 0.557·11-s − 0.279·13-s + 0.0847i·15-s + 0.341i·17-s − 1.05i·19-s + (−0.326 − 0.476i)21-s − 0.667i·23-s − 0.978·25-s + 0.192i·27-s + 0.377i·29-s − 0.723·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.332i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.943 + 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.323172908\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.323172908\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (-15.2 + 10.4i)T \) |
good | 5 | \( 1 + 1.64T + 125T^{2} \) |
| 11 | \( 1 - 20.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 13.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 23.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 87.7iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 73.6iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 58.9iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 124.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 56.5iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 135. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 259.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 217.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 529. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 685. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 149.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 409.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 885. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 269. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 902. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 623. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 986. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 179. iT - 9.12e5T^{2} \) |
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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
−8.755466334630712370988664938804, −7.998855966770420476033489688527, −7.26033575640780255790578496591, −6.59685278283308206436524792560, −5.53885780357024524074461386585, −4.58993087020881140596508276319, −3.73571732156766559622738133660, −2.41160212242501898163332463044, −1.41816002923306917944789299050, −0.30254989042383959049462529628,
1.38138147138962615894910462394, 2.47786164432120682046068641438, 3.72017432842385870374007172298, 4.44018614461051871888466334701, 5.48106046222824985865177274789, 6.02801769195980992650814672585, 7.39431048784019116940639587500, 7.957413696661820754545742673932, 8.975808005484646165056876780354, 9.436449801864359562664691923497