L(s) = 1 | + (−2.23 − 1.28i)2-s + (2.32 + 4.02i)4-s + (−1.16 − 2.01i)5-s − 6.82i·8-s + 6.01i·10-s + (3.78 + 2.18i)11-s + (−1.14 + 0.660i)13-s + (−4.15 + 7.18i)16-s − 5.78·17-s + 0.675i·19-s + (5.41 − 9.38i)20-s + (−5.62 − 9.74i)22-s + (4.81 − 2.78i)23-s + (−0.218 + 0.379i)25-s + 3.40·26-s + ⋯ |
L(s) = 1 | + (−1.57 − 0.911i)2-s + (1.16 + 2.01i)4-s + (−0.521 − 0.903i)5-s − 2.41i·8-s + 1.90i·10-s + (1.14 + 0.658i)11-s + (−0.317 + 0.183i)13-s + (−1.03 + 1.79i)16-s − 1.40·17-s + 0.154i·19-s + (1.21 − 2.09i)20-s + (−1.19 − 2.07i)22-s + (1.00 − 0.580i)23-s + (−0.0437 + 0.0758i)25-s + 0.667·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5244413507\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5244413507\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.23 + 1.28i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (1.16 + 2.01i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.78 - 2.18i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.14 - 0.660i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 5.78T + 17T^{2} \) |
| 19 | \( 1 - 0.675iT - 19T^{2} \) |
| 23 | \( 1 + (-4.81 + 2.78i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.86 + 2.23i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.47 + 2.00i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 3.01T + 37T^{2} \) |
| 41 | \( 1 + (-3.29 - 5.70i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.89 + 6.74i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.246 - 0.427i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 4.14iT - 53T^{2} \) |
| 59 | \( 1 + (2.15 + 3.73i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.77 - 1.02i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.41 - 4.17i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 1.17iT - 71T^{2} \) |
| 73 | \( 1 + 15.1iT - 73T^{2} \) |
| 79 | \( 1 + (-5.30 + 9.18i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.32 + 9.22i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 3.32T + 89T^{2} \) |
| 97 | \( 1 + (12.7 + 7.36i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.140842462796378452981783265841, −8.882175582804976250707538863492, −7.993071769164859968848209236384, −7.20655904690347963478756225985, −6.40032891311720249882842884827, −4.65454639698981507615498273335, −4.00971544782854197274837713878, −2.62883814085162643535312993994, −1.60172594573548004686061348392, −0.44963755344226189185047714538,
1.08595064753233438825264125580, 2.56123354072555847352485926458, 3.86091283606483411092336386091, 5.29199142966289045296368686008, 6.40648619446813024000988085981, 6.82234851265331932248300217134, 7.49592563159043959874059480249, 8.361550092605538425096560066253, 9.123291110042119755697779275391, 9.569189593628404241322994163920