L(s) = 1 | + (−1.80 + 1.04i)2-s + (1.17 − 2.03i)4-s + (−1.65 + 2.86i)5-s + 0.717i·8-s − 6.88i·10-s + (−2.30 + 1.33i)11-s + (2.11 + 1.21i)13-s + (1.59 + 2.76i)16-s + 7.18·17-s + 4.90i·19-s + (3.87 + 6.70i)20-s + (2.77 − 4.80i)22-s + (4.32 + 2.49i)23-s + (−2.96 − 5.12i)25-s − 5.08·26-s + ⋯ |
L(s) = 1 | + (−1.27 + 0.736i)2-s + (0.586 − 1.01i)4-s + (−0.738 + 1.27i)5-s + 0.253i·8-s − 2.17i·10-s + (−0.694 + 0.401i)11-s + (0.585 + 0.338i)13-s + (0.399 + 0.691i)16-s + 1.74·17-s + 1.12i·19-s + (0.866 + 1.50i)20-s + (0.591 − 1.02i)22-s + (0.901 + 0.520i)23-s + (−0.592 − 1.02i)25-s − 0.997·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 + 0.338i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.940 + 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5324675804\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5324675804\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (1.80 - 1.04i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (1.65 - 2.86i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.30 - 1.33i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.11 - 1.21i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 7.18T + 17T^{2} \) |
| 19 | \( 1 - 4.90iT - 19T^{2} \) |
| 23 | \( 1 + (-4.32 - 2.49i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (5.50 - 3.17i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.30 - 1.33i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 1.68T + 37T^{2} \) |
| 41 | \( 1 + (0.553 - 0.958i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.93 - 5.08i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.44 + 4.22i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 10.3iT - 53T^{2} \) |
| 59 | \( 1 + (-2.56 + 4.44i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.44 + 2.56i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.16 - 7.21i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 2.07iT - 71T^{2} \) |
| 73 | \( 1 + 8.01iT - 73T^{2} \) |
| 79 | \( 1 + (2.50 + 4.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.04 + 1.80i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 1.08T + 89T^{2} \) |
| 97 | \( 1 + (9.47 - 5.46i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.06120761564399335675768598306, −9.229673740350648614161150526696, −8.203591744589048323539840436344, −7.62506737983026908255406278987, −7.19558427216369764918918820402, −6.29871035146535229127384268059, −5.42407012265576420648501171401, −3.82213010401676098210051428520, −3.09115982348843135523112334565, −1.41205870081321018370630389341,
0.42039564784248118229084659792, 1.18424278584007977309694124860, 2.68327093233402812418523087982, 3.70698049052329629971539255783, 4.98751461389180809903541223028, 5.62467661700948038812511416423, 7.22465023415221602314504120133, 8.017712467625516440405421094089, 8.441930979765041758202299568285, 9.138719812859900186206908474642