L(s) = 1 | + (−1.40 − 0.112i)2-s + (−0.707 + 0.707i)3-s + (1.97 + 0.316i)4-s + (2.18 + 0.466i)5-s + (1.07 − 0.917i)6-s + 1.00·7-s + (−2.74 − 0.667i)8-s − 1.00i·9-s + (−3.03 − 0.903i)10-s + (−1.89 + 1.89i)11-s + (−1.61 + 1.17i)12-s + (2.65 − 2.65i)13-s + (−1.40 − 0.112i)14-s + (−1.87 + 1.21i)15-s + (3.80 + 1.24i)16-s + 1.73i·17-s + ⋯ |
L(s) = 1 | + (−0.996 − 0.0792i)2-s + (−0.408 + 0.408i)3-s + (0.987 + 0.158i)4-s + (0.977 + 0.208i)5-s + (0.439 − 0.374i)6-s + 0.378·7-s + (−0.971 − 0.235i)8-s − 0.333i·9-s + (−0.958 − 0.285i)10-s + (−0.571 + 0.571i)11-s + (−0.467 + 0.338i)12-s + (0.737 − 0.737i)13-s + (−0.376 − 0.0299i)14-s + (−0.484 + 0.314i)15-s + (0.950 + 0.312i)16-s + 0.421i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.826478 + 0.278622i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.826478 + 0.278622i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.40 + 0.112i)T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (-2.18 - 0.466i)T \) |
good | 7 | \( 1 - 1.00T + 7T^{2} \) |
| 11 | \( 1 + (1.89 - 1.89i)T - 11iT^{2} \) |
| 13 | \( 1 + (-2.65 + 2.65i)T - 13iT^{2} \) |
| 17 | \( 1 - 1.73iT - 17T^{2} \) |
| 19 | \( 1 + (-5.33 - 5.33i)T + 19iT^{2} \) |
| 23 | \( 1 - 0.160T + 23T^{2} \) |
| 29 | \( 1 + (-2.70 - 2.70i)T + 29iT^{2} \) |
| 31 | \( 1 + 4.64T + 31T^{2} \) |
| 37 | \( 1 + (-5.35 - 5.35i)T + 37iT^{2} \) |
| 41 | \( 1 + 9.89iT - 41T^{2} \) |
| 43 | \( 1 + (7.23 + 7.23i)T + 43iT^{2} \) |
| 47 | \( 1 - 4.79iT - 47T^{2} \) |
| 53 | \( 1 + (-3.44 - 3.44i)T + 53iT^{2} \) |
| 59 | \( 1 + (-0.101 + 0.101i)T - 59iT^{2} \) |
| 61 | \( 1 + (6.01 + 6.01i)T + 61iT^{2} \) |
| 67 | \( 1 + (-9.04 + 9.04i)T - 67iT^{2} \) |
| 71 | \( 1 + 4.60iT - 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 + 5.73T + 79T^{2} \) |
| 83 | \( 1 + (-2.04 + 2.04i)T - 83iT^{2} \) |
| 89 | \( 1 + 15.0iT - 89T^{2} \) |
| 97 | \( 1 - 3.84iT - 97T^{2} \) |
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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
−12.01023024581039377845573767287, −10.84989454396831462983827867370, −10.30437772534650940933515131561, −9.559592385256123891683301113185, −8.438488831201799242304837741023, −7.39239105107162476699486412674, −6.12822768198815203942713363949, −5.30032768437172517343143122757, −3.26799560049622987407066165164, −1.61054695647456507279130295443,
1.20564710733479193818265774696, 2.68182017553030303710997864705, 5.12116972476764387005721173790, 6.10801149843328458665334711879, 7.05091869783261290537604153386, 8.196382333645552857260715111658, 9.152728255298204408220934420944, 9.980801027305741319598377890843, 11.18707171739499807480840278321, 11.55189217014452018485210925718