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
−9.138719812859900186206908474642, −8.441930979765041758202299568285, −8.017712467625516440405421094089, −7.22465023415221602314504120133, −5.62467661700948038812511416423, −4.98751461389180809903541223028, −3.70698049052329629971539255783, −2.68327093233402812418523087982, −1.18424278584007977309694124860, −0.42039564784248118229084659792,
1.41205870081321018370630389341, 3.09115982348843135523112334565, 3.82213010401676098210051428520, 5.42407012265576420648501171401, 6.29871035146535229127384268059, 7.19558427216369764918918820402, 7.62506737983026908255406278987, 8.203591744589048323539840436344, 9.229673740350648614161150526696, 10.06120761564399335675768598306