L(s) = 1 | + (0.707 + 0.707i)2-s + (−1.58 − 0.707i)3-s + 1.00i·4-s + (−0.618 − 1.61i)6-s + (−0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s + (2.00 + 2.23i)9-s − 1.23i·11-s + (0.707 − 1.58i)12-s + (−1.74 − 1.74i)13-s − 1.00·14-s − 1.00·16-s + (−1.41 − 1.41i)17-s + (−0.166 + 2.99i)18-s + 5.23i·19-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (−0.912 − 0.408i)3-s + 0.500i·4-s + (−0.252 − 0.660i)6-s + (−0.267 + 0.267i)7-s + (−0.250 + 0.250i)8-s + (0.666 + 0.745i)9-s − 0.372i·11-s + (0.204 − 0.456i)12-s + (−0.484 − 0.484i)13-s − 0.267·14-s − 0.250·16-s + (−0.342 − 0.342i)17-s + (−0.0393 + 0.706i)18-s + 1.20i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3011631420\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3011631420\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (1.58 + 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 11 | \( 1 + 1.23iT - 11T^{2} \) |
| 13 | \( 1 + (1.74 + 1.74i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.41 + 1.41i)T + 17iT^{2} \) |
| 19 | \( 1 - 5.23iT - 19T^{2} \) |
| 23 | \( 1 + (4.57 - 4.57i)T - 23iT^{2} \) |
| 29 | \( 1 + 5.23T + 29T^{2} \) |
| 31 | \( 1 - 0.763T + 31T^{2} \) |
| 37 | \( 1 + (1.95 - 1.95i)T - 37iT^{2} \) |
| 41 | \( 1 + 4.76iT - 41T^{2} \) |
| 43 | \( 1 + (0.874 + 0.874i)T + 43iT^{2} \) |
| 47 | \( 1 + (8.61 + 8.61i)T + 47iT^{2} \) |
| 53 | \( 1 + (4.91 - 4.91i)T - 53iT^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 - 4.94T + 61T^{2} \) |
| 67 | \( 1 + (9.35 - 9.35i)T - 67iT^{2} \) |
| 71 | \( 1 - 15.4iT - 71T^{2} \) |
| 73 | \( 1 + (6.86 + 6.86i)T + 73iT^{2} \) |
| 79 | \( 1 - 4.94iT - 79T^{2} \) |
| 83 | \( 1 + (1.95 - 1.95i)T - 83iT^{2} \) |
| 89 | \( 1 + 4.76T + 89T^{2} \) |
| 97 | \( 1 + (-5.78 + 5.78i)T - 97iT^{2} \) |
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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
−10.36733590799524154446433153995, −9.640444878127869792533225393976, −8.415721667137602058261010890780, −7.62337082890045544247716477981, −6.90953097122937030317626052576, −5.85141177363834207166161759907, −5.55589095975755607283382483145, −4.42248701126860740966365060264, −3.31115845460641395005949076788, −1.84169251230660354208071275303,
0.12195826207085402574685112126, 1.84123788760882676365365835728, 3.24709796797310013658675335816, 4.44656062243461569306699047157, 4.78540435800776294196101748151, 6.06974914714461981677604019736, 6.62076051492912868366894022533, 7.61547866062942146776126646040, 9.054596628418384565679674326449, 9.689023942336153926508282491609