L(s) = 1 | + (0.780 + 1.17i)2-s − 3-s + (−0.780 + 1.84i)4-s + 1.69i·5-s + (−0.780 − 1.17i)6-s + (2.56 − 0.662i)7-s + (−2.78 + 0.516i)8-s + 9-s + (−2 + 1.32i)10-s − 3.02i·11-s + (0.780 − 1.84i)12-s − 6.04i·13-s + (2.78 + 2.50i)14-s − 1.69i·15-s + (−2.78 − 2.87i)16-s + 4.34i·17-s + ⋯ |
L(s) = 1 | + (0.552 + 0.833i)2-s − 0.577·3-s + (−0.390 + 0.920i)4-s + 0.758i·5-s + (−0.318 − 0.481i)6-s + (0.968 − 0.250i)7-s + (−0.983 + 0.182i)8-s + 0.333·9-s + (−0.632 + 0.418i)10-s − 0.910i·11-s + (0.225 − 0.531i)12-s − 1.67i·13-s + (0.743 + 0.669i)14-s − 0.437i·15-s + (−0.695 − 0.718i)16-s + 1.05i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.147 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.147 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.812383 + 0.700178i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.812383 + 0.700178i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.780 - 1.17i)T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + (-2.56 + 0.662i)T \) |
good | 5 | \( 1 - 1.69iT - 5T^{2} \) |
| 11 | \( 1 + 3.02iT - 11T^{2} \) |
| 13 | \( 1 + 6.04iT - 13T^{2} \) |
| 17 | \( 1 - 4.34iT - 17T^{2} \) |
| 19 | \( 1 + 1.12T + 19T^{2} \) |
| 23 | \( 1 - 3.02iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 7.12T + 37T^{2} \) |
| 41 | \( 1 + 7.73iT - 41T^{2} \) |
| 43 | \( 1 - 8.10iT - 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 9.43iT - 61T^{2} \) |
| 67 | \( 1 + 2.06iT - 67T^{2} \) |
| 71 | \( 1 + 12.4iT - 71T^{2} \) |
| 73 | \( 1 - 3.39iT - 73T^{2} \) |
| 79 | \( 1 + 4.71iT - 79T^{2} \) |
| 83 | \( 1 - 6.24T + 83T^{2} \) |
| 89 | \( 1 - 7.73iT - 89T^{2} \) |
| 97 | \( 1 - 8.68iT - 97T^{2} \) |
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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
−14.70794632546934691133953981860, −13.59521876635024976861823304341, −12.57979931575616458530725098471, −11.27063529803585012307819614926, −10.46670276775959189710085022577, −8.456778002023107639474112546269, −7.53555696829086494453008678384, −6.17918897770837174598230684376, −5.17481636866346656919955845781, −3.46753522626254578920840203411,
1.82160855892319050731105368928, 4.45884795055782048674184797584, 5.09078777665327975728685755752, 6.80710732998294700497151854885, 8.750367000463082385743015748634, 9.767264125272888931018523878461, 11.17435543798253164693965113176, 11.87880541162597413123835085301, 12.67746438077370435626351681081, 13.93241458532731957908340467952