L(s) = 1 | − 0.122i·2-s + 1.98·4-s + (−0.264 − 0.458i)5-s − 0.487i·8-s + (−0.0560 + 0.0323i)10-s + (−3.64 − 2.10i)11-s + (1.74 + 1.00i)13-s + 3.91·16-s + (−2.19 − 3.79i)17-s + (−4.54 − 2.62i)19-s + (−0.525 − 0.910i)20-s + (−0.257 + 0.445i)22-s + (5.43 − 3.13i)23-s + (2.35 − 4.08i)25-s + (0.123 − 0.213i)26-s + ⋯ |
L(s) = 1 | − 0.0865i·2-s + 0.992·4-s + (−0.118 − 0.205i)5-s − 0.172i·8-s + (−0.0177 + 0.0102i)10-s + (−1.09 − 0.633i)11-s + (0.484 + 0.279i)13-s + 0.977·16-s + (−0.532 − 0.921i)17-s + (−1.04 − 0.601i)19-s + (−0.117 − 0.203i)20-s + (−0.0548 + 0.0949i)22-s + (1.13 − 0.654i)23-s + (0.471 − 0.817i)25-s + (0.0242 − 0.0419i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.310 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.844794108\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.844794108\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.122iT - 2T^{2} \) |
| 5 | \( 1 + (0.264 + 0.458i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (3.64 + 2.10i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.74 - 1.00i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.19 + 3.79i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.54 + 2.62i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.43 + 3.13i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-7.27 + 4.20i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 1.19iT - 31T^{2} \) |
| 37 | \( 1 + (-1.61 + 2.79i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.0994 - 0.172i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.96 - 6.86i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 9.97T + 47T^{2} \) |
| 53 | \( 1 + (3.65 - 2.10i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 + 13.1iT - 61T^{2} \) |
| 67 | \( 1 + 6.58T + 67T^{2} \) |
| 71 | \( 1 - 8.50iT - 71T^{2} \) |
| 73 | \( 1 + (-4.86 + 2.80i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 0.572T + 79T^{2} \) |
| 83 | \( 1 + (-5.42 - 9.39i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (6.43 - 11.1i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.493 - 0.285i)T + (48.5 - 84.0i)T^{2} \) |
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
−9.498748744273191568581468574567, −8.463061689715735618389479176437, −7.995856630716207116227576987674, −6.76128499130003942092976456418, −6.44994649320466551310535868476, −5.24056459757554536663009331540, −4.39172091083855443083282470310, −2.95671789504805843482240560106, −2.40831181383163848691650242653, −0.74958101896502099700472357145,
1.54252830425411841215694674605, 2.61606666010297532875815303689, 3.53215974500869156925861418616, 4.81092158731867556720935788026, 5.73961756754339881414607746850, 6.60869148063559315815463868234, 7.26445182353466276222017986918, 8.111418921728846158123604694372, 8.785219052260771077045530291892, 10.11967360211260506455900925326