L(s) = 1 | + (1.48 − 1.48i)3-s + (−1.83 − 1.83i)5-s − i·7-s − 1.40i·9-s + (0.321 + 0.321i)11-s + (−4.61 + 4.61i)13-s − 5.45·15-s − 1.84·17-s + (−3.88 + 3.88i)19-s + (−1.48 − 1.48i)21-s − 5.88i·23-s + 1.74i·25-s + (2.36 + 2.36i)27-s + (−6.14 + 6.14i)29-s − 5.69·31-s + ⋯ |
L(s) = 1 | + (0.857 − 0.857i)3-s + (−0.821 − 0.821i)5-s − 0.377i·7-s − 0.469i·9-s + (0.0969 + 0.0969i)11-s + (−1.28 + 1.28i)13-s − 1.40·15-s − 0.446·17-s + (−0.892 + 0.892i)19-s + (−0.323 − 0.323i)21-s − 1.22i·23-s + 0.348i·25-s + (0.454 + 0.454i)27-s + (−1.14 + 1.14i)29-s − 1.02·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1417563335\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1417563335\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + (-1.48 + 1.48i)T - 3iT^{2} \) |
| 5 | \( 1 + (1.83 + 1.83i)T + 5iT^{2} \) |
| 11 | \( 1 + (-0.321 - 0.321i)T + 11iT^{2} \) |
| 13 | \( 1 + (4.61 - 4.61i)T - 13iT^{2} \) |
| 17 | \( 1 + 1.84T + 17T^{2} \) |
| 19 | \( 1 + (3.88 - 3.88i)T - 19iT^{2} \) |
| 23 | \( 1 + 5.88iT - 23T^{2} \) |
| 29 | \( 1 + (6.14 - 6.14i)T - 29iT^{2} \) |
| 31 | \( 1 + 5.69T + 31T^{2} \) |
| 37 | \( 1 + (1.66 + 1.66i)T + 37iT^{2} \) |
| 41 | \( 1 + 10.7iT - 41T^{2} \) |
| 43 | \( 1 + (0.533 + 0.533i)T + 43iT^{2} \) |
| 47 | \( 1 - 0.465T + 47T^{2} \) |
| 53 | \( 1 + (-0.623 - 0.623i)T + 53iT^{2} \) |
| 59 | \( 1 + (-7.32 - 7.32i)T + 59iT^{2} \) |
| 61 | \( 1 + (-7.57 + 7.57i)T - 61iT^{2} \) |
| 67 | \( 1 + (6.16 - 6.16i)T - 67iT^{2} \) |
| 71 | \( 1 - 0.162iT - 71T^{2} \) |
| 73 | \( 1 - 3.49iT - 73T^{2} \) |
| 79 | \( 1 + 8.28T + 79T^{2} \) |
| 83 | \( 1 + (2.51 - 2.51i)T - 83iT^{2} \) |
| 89 | \( 1 - 1.60iT - 89T^{2} \) |
| 97 | \( 1 + 8.88T + 97T^{2} \) |
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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
−8.709296028139330995366347554516, −8.040934338836710038116007289777, −7.08730355919623061201765573771, −6.95412212644063940323070054509, −5.41959556309951657151735004161, −4.39347461287624733759072922159, −3.84947941962873420756934712530, −2.39611135342487877444216618006, −1.66470528622489210226352309317, −0.04328435743199586819985405425,
2.34483717812596202309186349115, 3.08677305782644115657005910229, 3.78547897277858658056944881883, 4.69207603048192313544162060246, 5.64813230261608566092950247301, 6.81085695167444766605229277889, 7.60279359498119487197190530873, 8.177789288969568322192653522324, 9.112044788092190701741387265543, 9.702221944158324001216266464579