L(s) = 1 | + (−0.866 − 1.5i)3-s + (−0.5 + 0.866i)5-s + (−1 + 1.73i)9-s + 1.73·15-s + (0.866 − 1.5i)23-s + (−0.499 − 0.866i)25-s + 1.73·27-s − 29-s − 41-s − 1.73·43-s + (−1 − 1.73i)45-s + (0.5 − 0.866i)61-s + (−0.866 − 1.5i)67-s − 3·69-s + (−0.866 + 1.49i)75-s + ⋯ |
L(s) = 1 | + (−0.866 − 1.5i)3-s + (−0.5 + 0.866i)5-s + (−1 + 1.73i)9-s + 1.73·15-s + (0.866 − 1.5i)23-s + (−0.499 − 0.866i)25-s + 1.73·27-s − 29-s − 41-s − 1.73·43-s + (−1 − 1.73i)45-s + (0.5 − 0.866i)61-s + (−0.866 − 1.5i)67-s − 3·69-s + (−0.866 + 1.49i)75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2919152238\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2919152238\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 + 1.73T + T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + 1.73T + T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.062781919348673036920044042519, −7.32273025709275823153683993186, −6.76888266107113899432176413754, −6.36595439830101845381323828970, −5.47604777892725539081538877560, −4.64371803828730038681310430530, −3.42626318057116166809757937077, −2.51428209346487921676847810688, −1.58503328259013118929370453294, −0.19106114900181863711018993316,
1.40622152745667324420238622059, 3.18696721227994827156518479338, 3.85817765665858858602347762403, 4.53442204220787829026252318553, 5.37018182768445738903213054815, 5.54733266871988576128343171652, 6.74396975215330874521149283845, 7.59908785334340121379329085271, 8.581054020172310062149742119351, 9.097987958290444683545724153719