L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s − 5-s + 0.999·8-s + (0.5 − 0.866i)10-s + (−0.5 + 0.866i)16-s + 1.73i·17-s + 19-s + (0.499 + 0.866i)20-s + 23-s + 25-s + 1.73i·31-s + (−0.499 − 0.866i)32-s + (−1.49 − 0.866i)34-s + (−0.5 + 0.866i)38-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s − 5-s + 0.999·8-s + (0.5 − 0.866i)10-s + (−0.5 + 0.866i)16-s + 1.73i·17-s + 19-s + (0.499 + 0.866i)20-s + 23-s + 25-s + 1.73i·31-s + (−0.499 − 0.866i)32-s + (−1.49 − 0.866i)34-s + (−0.5 + 0.866i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6300854967\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6300854967\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 - 1.73iT - T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.73iT - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 2T + T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - 1.73iT - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 1.73iT - T^{2} \) |
| 83 | \( 1 + 1.73iT - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 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
−10.35926863981332916197547021777, −9.115339151981077370900375116296, −8.598600174486110832441660758292, −7.75015773205395959556155777485, −7.13248977114332854602833941458, −6.23126602560062868844363845072, −5.22912652763522422513127345830, −4.32950983419330365907307290421, −3.27585126106885112450264898127, −1.32711699299013869916523693145,
0.814848633076077728505187869397, 2.58373158749381746898024116209, 3.41192639864147149906437493751, 4.43440647885073440377577467162, 5.26731308621504286182571085655, 6.93718049286328161581528628275, 7.55358607429323183828325448134, 8.256761610070114094721929193796, 9.325607641213754934793097843353, 9.660394863117190605355211842937