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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0868i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.02135183617\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02135183617\) |
\(L(\frac{5}{2})\) |
|
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
−8.954642170435430727233616790887, −8.052466314581413952246932394961, −7.04160912103423848558096862763, −6.25165868067078201441845121849, −5.69981393870740356878420204052, −4.78905506510578620034752830602, −3.33742629215002672841190178939, −2.23983633656523297153183146297, −1.68808869899923630828908458598, −0.00433674185245126589541753958,
1.41584882816721148072420387846, 2.61368447710084743882217206902, 3.60870053917279667190656826836, 4.71732578905697948254566673507, 5.31858957727880658245944243424, 6.48668328251658784990572401403, 7.00565019680415338179315458564, 8.064550648216045503138368764495, 9.262305744474746621782156647198, 9.643564488300076437327716762116