L(s) = 1 | + (0.5 + 0.866i)3-s + (−2.78 − 1.60i)5-s + (−0.499 + 0.866i)9-s + (−1.18 + 0.683i)11-s − 2.93i·13-s − 3.21i·15-s + (5.98 − 3.45i)17-s + (−3.67 + 6.36i)19-s + (3.14 + 1.81i)23-s + (2.66 + 4.62i)25-s − 0.999·27-s − 1.11·29-s + (−4.35 − 7.53i)31-s + (−1.18 − 0.683i)33-s + (−3.81 + 6.61i)37-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (−1.24 − 0.718i)5-s + (−0.166 + 0.288i)9-s + (−0.356 + 0.206i)11-s − 0.812i·13-s − 0.830i·15-s + (1.45 − 0.838i)17-s + (−0.843 + 1.46i)19-s + (0.655 + 0.378i)23-s + (0.533 + 0.924i)25-s − 0.192·27-s − 0.206·29-s + (−0.781 − 1.35i)31-s + (−0.206 − 0.118i)33-s + (−0.627 + 1.08i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6299396600\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6299396600\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (2.78 + 1.60i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.18 - 0.683i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2.93iT - 13T^{2} \) |
| 17 | \( 1 + (-5.98 + 3.45i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.67 - 6.36i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.14 - 1.81i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 1.11T + 29T^{2} \) |
| 31 | \( 1 + (4.35 + 7.53i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.81 - 6.61i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 0.833iT - 41T^{2} \) |
| 43 | \( 1 - 4.82iT - 43T^{2} \) |
| 47 | \( 1 + (1.47 - 2.55i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.28 - 3.96i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.04 - 12.2i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (9.57 + 5.53i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (10.5 - 6.07i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12.6iT - 71T^{2} \) |
| 73 | \( 1 + (5.62 - 3.24i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.74 + 3.89i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 8.87T + 83T^{2} \) |
| 89 | \( 1 + (10.9 + 6.30i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.178361663049521390566095560843, −8.368118582386281785730566055519, −7.78768201783166581695638480830, −7.38152492265526191979902044550, −5.88258334530173295019799386573, −5.23243915225514765767996249015, −4.34868343366538165232429624949, −3.64763039152111577822026148875, −2.83118087129304447205968351866, −1.20991554070118204775064780632,
0.22418125345043180118568990564, 1.80870915708575117478854993229, 3.02337912902322873157620390414, 3.60840803564348742470839652341, 4.57213673569601926441240226038, 5.62985491368730198540113754090, 6.78371542704424068823263923305, 7.09204891490600170879717500774, 7.891033935077345687139384686885, 8.594269363192095680897799800635