L(s) = 1 | + (0.555 + 0.831i)2-s + (1.17 + 1.17i)3-s + (−0.382 + 0.923i)4-s + (−0.707 + 0.707i)5-s + (−0.324 + 1.63i)6-s + (−0.980 + 0.195i)8-s + 1.76i·9-s + (−0.980 − 0.195i)10-s + (1.30 − 1.30i)11-s + (−1.53 + 0.636i)12-s + (0.275 + 0.275i)13-s − 1.66·15-s + (−0.707 − 0.707i)16-s + (−1.46 + 0.980i)18-s + (−0.707 − 0.707i)19-s + (−0.382 − 0.923i)20-s + ⋯ |
L(s) = 1 | + (0.555 + 0.831i)2-s + (1.17 + 1.17i)3-s + (−0.382 + 0.923i)4-s + (−0.707 + 0.707i)5-s + (−0.324 + 1.63i)6-s + (−0.980 + 0.195i)8-s + 1.76i·9-s + (−0.980 − 0.195i)10-s + (1.30 − 1.30i)11-s + (−1.53 + 0.636i)12-s + (0.275 + 0.275i)13-s − 1.66·15-s + (−0.707 − 0.707i)16-s + (−1.46 + 0.980i)18-s + (−0.707 − 0.707i)19-s + (−0.382 − 0.923i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.856810817\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.856810817\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.555 - 0.831i)T \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| 19 | \( 1 + (0.707 + 0.707i)T \) |
good | 3 | \( 1 + (-1.17 - 1.17i)T + iT^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + (-1.30 + 1.30i)T - iT^{2} \) |
| 13 | \( 1 + (-0.275 - 0.275i)T + iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-0.785 + 0.785i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (1.17 - 1.17i)T - iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + (1.30 + 1.30i)T + iT^{2} \) |
| 67 | \( 1 + (-1.38 - 1.38i)T + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - 0.390T + T^{2} \) |
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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
−9.633397266985026204424250419941, −9.006567868767083215550311658118, −8.433012135885577094342925263722, −7.77848618535226382166838037810, −6.72188011443680063394828955847, −6.05733956531544106668549624511, −4.69922095691067854345299879846, −3.99362767338732329713116895473, −3.46769695301328532155099583705, −2.67844536467189548595527954524,
1.29956655539754323553454123186, 1.94636984230947020815826415638, 3.20757414501283725303520060110, 4.00068213838531930773924164125, 4.76569321820198292471749405155, 6.19539643687500308563140899512, 6.90338166613658165530413798471, 7.88140365186797646275248367663, 8.521914081671132702861571599236, 9.297881243767336534735543798227