L(s) = 1 | + (0.707 − 0.707i)3-s + (−0.334 − 0.334i)5-s − 4.55i·7-s − 1.00i·9-s + (2.47 + 2.47i)11-s + (−0.0594 + 0.0594i)13-s − 0.473·15-s + 3.61·17-s + (−2.55 + 2.55i)19-s + (−3.22 − 3.22i)21-s + 2.82i·23-s − 4.77i·25-s + (−0.707 − 0.707i)27-s + (−5.16 + 5.16i)29-s + 0.557·31-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (−0.149 − 0.149i)5-s − 1.72i·7-s − 0.333i·9-s + (0.745 + 0.745i)11-s + (−0.0164 + 0.0164i)13-s − 0.122·15-s + 0.877·17-s + (−0.586 + 0.586i)19-s + (−0.703 − 0.703i)21-s + 0.589i·23-s − 0.955i·25-s + (−0.136 − 0.136i)27-s + (−0.958 + 0.958i)29-s + 0.100·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.589 + 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.589 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.17409 - 0.596936i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.17409 - 0.596936i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
good | 5 | \( 1 + (0.334 + 0.334i)T + 5iT^{2} \) |
| 7 | \( 1 + 4.55iT - 7T^{2} \) |
| 11 | \( 1 + (-2.47 - 2.47i)T + 11iT^{2} \) |
| 13 | \( 1 + (0.0594 - 0.0594i)T - 13iT^{2} \) |
| 17 | \( 1 - 3.61T + 17T^{2} \) |
| 19 | \( 1 + (2.55 - 2.55i)T - 19iT^{2} \) |
| 23 | \( 1 - 2.82iT - 23T^{2} \) |
| 29 | \( 1 + (5.16 - 5.16i)T - 29iT^{2} \) |
| 31 | \( 1 - 0.557T + 31T^{2} \) |
| 37 | \( 1 + (-4.38 - 4.38i)T + 37iT^{2} \) |
| 41 | \( 1 - 9.27iT - 41T^{2} \) |
| 43 | \( 1 + (-1.61 - 1.61i)T + 43iT^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + (0.493 + 0.493i)T + 53iT^{2} \) |
| 59 | \( 1 + (4 + 4i)T + 59iT^{2} \) |
| 61 | \( 1 + (-2.72 + 2.72i)T - 61iT^{2} \) |
| 67 | \( 1 + (3.77 - 3.77i)T - 67iT^{2} \) |
| 71 | \( 1 - 9.11iT - 71T^{2} \) |
| 73 | \( 1 - 0.541iT - 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + (-10.6 + 10.6i)T - 83iT^{2} \) |
| 89 | \( 1 + 14.6iT - 89T^{2} \) |
| 97 | \( 1 - 4.31T + 97T^{2} \) |
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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
−12.54719163938478274563375368570, −11.46897971494646271760803602847, −10.30337195795278484303486004226, −9.538521801529334999233302943317, −8.100920191570925532660881592478, −7.35939443846833580462171004812, −6.39483443843158240436950094343, −4.53959882077880813622089041202, −3.53948424431201615160273826919, −1.38432255337464431038607124454,
2.39732776623493553556057084242, 3.69853591472143598701898606224, 5.30690829790213479006136316781, 6.24694480605942912739270865837, 7.82576401838318948361709634206, 8.912363983307604838953398953482, 9.368552449865590050342718848364, 10.82170796358521215287844549025, 11.72247067700653786803777008253, 12.55327591012681365086364136403