L(s) = 1 | + (0.959 − 1.66i)2-s + (−0.841 − 1.45i)4-s + (−1.43 − 2.49i)5-s + (2.39 + 4.15i)7-s + 0.609·8-s − 5.52·10-s + (−2.29 − 3.97i)11-s − 5.61·13-s + 9.20·14-s + (2.26 − 3.92i)16-s + (−2.90 − 5.02i)17-s + (4.10 − 7.10i)19-s + (−2.42 + 4.19i)20-s − 8.80·22-s − 1.59·23-s + ⋯ |
L(s) = 1 | + (0.678 − 1.17i)2-s + (−0.420 − 0.728i)4-s + (−0.643 − 1.11i)5-s + (0.906 + 1.57i)7-s + 0.215·8-s − 1.74·10-s + (−0.691 − 1.19i)11-s − 1.55·13-s + 2.46·14-s + (0.566 − 0.981i)16-s + (−0.703 − 1.21i)17-s + (0.940 − 1.62i)19-s + (−0.541 + 0.937i)20-s − 1.87·22-s − 0.333·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.276i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.259841 - 1.84100i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.259841 - 1.84100i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 103 | \( 1 + (-10.1 - 0.116i)T \) |
good | 2 | \( 1 + (-0.959 + 1.66i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (1.43 + 2.49i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-2.39 - 4.15i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.29 + 3.97i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 5.61T + 13T^{2} \) |
| 17 | \( 1 + (2.90 + 5.02i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.10 + 7.10i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 1.59T + 23T^{2} \) |
| 29 | \( 1 + (-1.07 + 1.86i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 0.283T + 31T^{2} \) |
| 37 | \( 1 - 9.23T + 37T^{2} \) |
| 41 | \( 1 + (3.46 - 5.99i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.04 + 5.27i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.34 - 2.33i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.95 - 5.12i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.59 - 2.76i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + 5.23T + 61T^{2} \) |
| 67 | \( 1 + (0.395 + 0.684i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.80 - 11.7i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 4.36T + 73T^{2} \) |
| 79 | \( 1 + 0.555T + 79T^{2} \) |
| 83 | \( 1 + (-0.116 + 0.201i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 6.40T + 89T^{2} \) |
| 97 | \( 1 + (-1.62 + 2.81i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.625458548851661262537378069603, −9.011103171370726415273832602919, −8.155211141942607381853291303924, −7.40290999953936756297519646876, −5.60350132266383814199843060449, −4.89745469896178361446718039929, −4.58866565537948004604613140819, −2.83248741571502648109215821991, −2.43703547452779180418694961211, −0.69690062427832067273420959995,
1.93387074128283271455849517414, 3.66044186687764846383531734522, 4.39421779908834847631176837743, 5.10409232589054370530892822276, 6.37852455640340755424016898063, 7.24367421936375421363838040231, 7.61599449368346499684378502962, 7.992914524338859006281782966469, 9.981594135686729426549515246352, 10.38100342356956695388074234554