L(s) = 1 | + 1.93i·2-s + (−0.991 + 0.130i)3-s − 2.73·4-s + (−0.252 − 1.91i)6-s − 3.34i·8-s + (0.965 − 0.258i)9-s + (2.70 − 0.356i)12-s + 1.58i·13-s + 3.73·16-s + (0.500 + 1.86i)18-s + i·23-s + (0.436 + 3.31i)24-s + 25-s − 3.06·26-s + (−0.923 + 0.382i)27-s + ⋯ |
L(s) = 1 | + 1.93i·2-s + (−0.991 + 0.130i)3-s − 2.73·4-s + (−0.252 − 1.91i)6-s − 3.34i·8-s + (0.965 − 0.258i)9-s + (2.70 − 0.356i)12-s + 1.58i·13-s + 3.73·16-s + (0.500 + 1.86i)18-s + i·23-s + (0.436 + 3.31i)24-s + 25-s − 3.06·26-s + (−0.923 + 0.382i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5011554375\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5011554375\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.991 - 0.130i)T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - iT \) |
good | 2 | \( 1 - 1.93iT - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - 1.58iT - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 29 | \( 1 - 1.73iT - T^{2} \) |
| 31 | \( 1 + 1.21iT - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.58T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 0.261T + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + 0.765T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + iT - T^{2} \) |
| 73 | \( 1 - 1.98iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 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
−9.192107495538065154263865725666, −8.450814312464050700626614636087, −7.47390743254484136717023753498, −6.89312799680698996652130193297, −6.51100764873944053992121565730, −5.63091977652730628586934607293, −5.03946493053968320209674156570, −4.38053959842891395401627192191, −3.60052290901899829051335503218, −1.38924801632748305815717444346,
0.39873672058022561966972632455, 1.39410064048710565057669692772, 2.54308069623074898826216183104, 3.32740762092355845891733893782, 4.32092150534841392735775015689, 4.99727631133845639374072426711, 5.61372576838449309554344262568, 6.62253624953037042246736499100, 7.85034465761565555622428401074, 8.443508041185664964046052956947