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 | \( 1 + T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.443508041185664964046052956947, −7.85034465761565555622428401074, −6.62253624953037042246736499100, −5.61372576838449309554344262568, −4.99727631133845639374072426711, −4.32092150534841392735775015689, −3.32740762092355845891733893782, −2.54308069623074898826216183104, −1.39410064048710565057669692772, −0.39873672058022561966972632455,
1.38924801632748305815717444346, 3.60052290901899829051335503218, 4.38053959842891395401627192191, 5.03946493053968320209674156570, 5.63091977652730628586934607293, 6.51100764873944053992121565730, 6.89312799680698996652130193297, 7.47390743254484136717023753498, 8.450814312464050700626614636087, 9.192107495538065154263865725666