L(s) = 1 | + 4.24i·7-s + 4·11-s − 6·13-s − 4.24i·17-s − 2.82i·19-s − 6·23-s + 5·25-s − 8.48i·29-s − 4.24i·31-s − 6·37-s + 1.41i·41-s − 2.82i·43-s + 6·47-s − 10.9·49-s − 8.48i·53-s + ⋯ |
L(s) = 1 | + 1.60i·7-s + 1.20·11-s − 1.66·13-s − 1.02i·17-s − 0.648i·19-s − 1.25·23-s + 25-s − 1.57i·29-s − 0.762i·31-s − 0.986·37-s + 0.220i·41-s − 0.431i·43-s + 0.875·47-s − 1.57·49-s − 1.16i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.383117932\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.383117932\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 4.24iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 6T + 13T^{2} \) |
| 17 | \( 1 + 4.24iT - 17T^{2} \) |
| 19 | \( 1 + 2.82iT - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 8.48iT - 29T^{2} \) |
| 31 | \( 1 + 4.24iT - 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 1.41iT - 41T^{2} \) |
| 43 | \( 1 + 2.82iT - 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 8.48iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 11.3iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 4.24iT - 79T^{2} \) |
| 83 | \( 1 - 16T + 83T^{2} \) |
| 89 | \( 1 - 12.7iT - 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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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
−8.281727875996340205926728840513, −7.47492027985152493306872111186, −6.69741620459850175635473403581, −6.05531711658694102965216739370, −5.18256315039191513160615503854, −4.67740197380450760806122119072, −3.59157795495890434429487960703, −2.45500455513515700810367104301, −2.15372077953322848092943999975, −0.42060422220265417722496646386,
1.03182485936910985551663084346, 1.88672077314071183445224226437, 3.24675181572872360970706542169, 3.95998620672111673457320435944, 4.53845935531958948573654704694, 5.43456788876945991080903544483, 6.50033308898321330160115165509, 7.00541818793748716783120293566, 7.54836601528391915874341346443, 8.379242369129545391056555770967