L(s) = 1 | + 0.193i·2-s + 1.96·4-s + (−1.48 − 1.67i)5-s + 0.768i·8-s + (0.324 − 0.287i)10-s − 2·11-s − 1.35i·13-s + 3.77·16-s + 3.35i·17-s + 5.35·19-s + (−2.90 − 3.28i)20-s − 0.387i·22-s − 4.96i·23-s + (−0.612 + 4.96i)25-s + 0.261·26-s + ⋯ |
L(s) = 1 | + 0.137i·2-s + 0.981·4-s + (−0.662 − 0.749i)5-s + 0.271i·8-s + (0.102 − 0.0908i)10-s − 0.603·11-s − 0.374i·13-s + 0.943·16-s + 0.812i·17-s + 1.22·19-s + (−0.649 − 0.735i)20-s − 0.0826i·22-s − 1.03i·23-s + (−0.122 + 0.992i)25-s + 0.0513·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.930065117\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.930065117\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.48 + 1.67i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 0.193iT - 2T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 1.35iT - 13T^{2} \) |
| 17 | \( 1 - 3.35iT - 17T^{2} \) |
| 19 | \( 1 - 5.35T + 19T^{2} \) |
| 23 | \( 1 + 4.96iT - 23T^{2} \) |
| 29 | \( 1 - 7.92T + 29T^{2} \) |
| 31 | \( 1 + 4.57T + 31T^{2} \) |
| 37 | \( 1 - 0.775iT - 37T^{2} \) |
| 41 | \( 1 - 3.73T + 41T^{2} \) |
| 43 | \( 1 + 12.6iT - 43T^{2} \) |
| 47 | \( 1 + 9.92iT - 47T^{2} \) |
| 53 | \( 1 + 8.57iT - 53T^{2} \) |
| 59 | \( 1 - 8.62T + 59T^{2} \) |
| 61 | \( 1 - 8.70T + 61T^{2} \) |
| 67 | \( 1 + 9.92iT - 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 9.35iT - 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 - 3.22iT - 83T^{2} \) |
| 89 | \( 1 + 1.03T + 89T^{2} \) |
| 97 | \( 1 + 18.4iT - 97T^{2} \) |
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
−8.507366467240182929316544239730, −8.329706066213262633141138523711, −7.34949446876726803716045961004, −6.80929842696206638744652595151, −5.64995651656405357007569668217, −5.16857235674011472516359668274, −3.99056743567381963463129263038, −3.10936214990072829714542304890, −2.03097240904092881299620156597, −0.73423012105103947301503000113,
1.18837237314975536193141472076, 2.65680067210576098649445646880, 3.05207443399162592910040032565, 4.14468843510134093452987759364, 5.26607360879130115612858565165, 6.15130262711706277137457540226, 6.99626748412049864643265360046, 7.52382664815572065367308823714, 8.060613781402334102507169459361, 9.322378737772725856476573490561