| L(s) = 1 | + (−1 − i)3-s + (−3 + 3i)7-s − i·9-s + 6·21-s + (1 + i)23-s + (−4 + 4i)27-s − 6i·29-s + 12·41-s + (9 + 9i)43-s + (7 − 7i)47-s − 11i·49-s + 8·61-s + (3 + 3i)63-s + (3 − 3i)67-s − 2i·69-s + ⋯ |
| L(s) = 1 | + (−0.577 − 0.577i)3-s + (−1.13 + 1.13i)7-s − 0.333i·9-s + 1.30·21-s + (0.208 + 0.208i)23-s + (−0.769 + 0.769i)27-s − 1.11i·29-s + 1.87·41-s + (1.37 + 1.37i)43-s + (1.02 − 1.02i)47-s − 1.57i·49-s + 1.02·61-s + (0.377 + 0.377i)63-s + (0.366 − 0.366i)67-s − 0.240i·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.039191406\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.039191406\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (1 + i)T + 3iT^{2} \) |
| 7 | \( 1 + (3 - 3i)T - 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 13iT^{2} \) |
| 17 | \( 1 + 17iT^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + (-1 - i)T + 23iT^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37iT^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 + (-9 - 9i)T + 43iT^{2} \) |
| 47 | \( 1 + (-7 + 7i)T - 47iT^{2} \) |
| 53 | \( 1 - 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + (-3 + 3i)T - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (11 + 11i)T + 83iT^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 + 97iT^{2} \) |
| 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
−9.378468980958517742437818312667, −8.676683686322070143775019249456, −7.58391356396562637697442744786, −6.79580121814763522725822710827, −5.92255289785252634124047291822, −5.73648311835224548715439573185, −4.29392024536477669311303045219, −3.17512173195372653193629663407, −2.24603293672649670424589853110, −0.66370069951966065907765649761,
0.78258933533769630609459500687, 2.56100977990067252553594394880, 3.74131552264969916001740768214, 4.35635188458020468337971909209, 5.38943781518919203046266981553, 6.18119996349557034845715173650, 7.09655167360036780894579923693, 7.67013848081013162138557875546, 8.908489870436310052046291335253, 9.604994132106875945435616786468
