L(s) = 1 | + (−2 − i)5-s + 2i·7-s − 6·11-s − 2i·13-s + 6i·17-s + 4·19-s − 8i·23-s + (3 + 4i)25-s − 8·31-s + (2 − 4i)35-s − 2i·37-s + 6·41-s + 4i·43-s + 4i·47-s + 3·49-s + ⋯ |
L(s) = 1 | + (−0.894 − 0.447i)5-s + 0.755i·7-s − 1.80·11-s − 0.554i·13-s + 1.45i·17-s + 0.917·19-s − 1.66i·23-s + (0.600 + 0.800i)25-s − 1.43·31-s + (0.338 − 0.676i)35-s − 0.328i·37-s + 0.937·41-s + 0.609i·43-s + 0.583i·47-s + 0.428·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.048373254\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.048373254\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−8.591232557839592344483951615885, −7.915480636398502045207174142754, −7.57583942966493404689729995986, −6.34241770203881660642586944179, −5.46614414165766877410124206058, −4.98578568565887513645239417793, −3.94249437456307424426351197869, −3.02565526879163106159855920807, −2.13164458407572418486467577150, −0.52156910168454150213925077464,
0.70908189711121200014294892689, 2.34776271499390363013391814930, 3.24659195737275121025427609762, 3.96700107970389237009371843789, 5.03722202144207354110376420757, 5.53072305748612743217766137095, 6.92391026891074056307535428500, 7.51680130803542416449981098982, 7.64725977729353684673565540597, 8.786103628151456489485243511870