L(s) = 1 | + (0.707 − 0.707i)2-s + (−1.84 − 0.765i)3-s − 1.00i·4-s + (−1.84 + 0.765i)6-s + (1.53 + 3.69i)7-s + (−0.707 − 0.707i)8-s + (0.707 + 0.707i)9-s + (−5.54 + 2.29i)11-s + (−0.765 + 1.84i)12-s + 2i·13-s + (3.69 + 1.53i)14-s − 1.00·16-s + 1.00·18-s + (−2.82 + 2.82i)19-s − 8i·21-s + (−2.29 + 5.54i)22-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−1.06 − 0.441i)3-s − 0.500i·4-s + (−0.754 + 0.312i)6-s + (0.578 + 1.39i)7-s + (−0.250 − 0.250i)8-s + (0.235 + 0.235i)9-s + (−1.67 + 0.692i)11-s + (−0.220 + 0.533i)12-s + 0.554i·13-s + (0.987 + 0.409i)14-s − 0.250·16-s + 0.235·18-s + (−0.648 + 0.648i)19-s − 1.74i·21-s + (−0.489 + 1.18i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.401 - 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.401 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.641672 + 0.419428i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.641672 + 0.419428i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 17 | \( 1 \) |
good | 3 | \( 1 + (1.84 + 0.765i)T + (2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (-3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (-1.53 - 3.69i)T + (-4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (5.54 - 2.29i)T + (7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 19 | \( 1 + (2.82 - 2.82i)T - 19iT^{2} \) |
| 23 | \( 1 + (16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + (-3.69 - 1.53i)T + (21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (3.69 + 1.53i)T + (26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (2.29 + 5.54i)T + (-28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (-5.65 - 5.65i)T + 43iT^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + (4.24 - 4.24i)T - 53iT^{2} \) |
| 59 | \( 1 + 59iT^{2} \) |
| 61 | \( 1 + (1.53 + 3.69i)T + (-43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (0.765 - 1.84i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (7.39 - 3.06i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 + (-5.35 + 12.9i)T + (-68.5 - 68.5i)T^{2} \) |
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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
−11.04074389495311970460689568521, −10.40542156863219522143966398183, −9.218364870110417322565994755427, −8.243832435611421550846046238413, −7.09175246482975238386761005775, −5.97903108282767474185210363705, −5.36080278227913339530160276258, −4.63880412886332289795380643043, −2.82079590106760747036820582231, −1.76978784547308270676853613027,
0.40538379771668143387815218752, 2.88239727090700862930137875159, 4.34578096171308629720812868309, 4.96520551232758261829069689480, 5.78719727389558325439045818790, 6.81151302833960658208461099521, 7.81158650792934785463246696561, 8.444079424570688176225644429382, 10.17649118179883685298641603079, 10.68437511609085395250061368592