| 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 | \( 1 - 97iT^{2} \) |
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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
−9.604994132106875945435616786468, −8.908489870436310052046291335253, −7.67013848081013162138557875546, −7.09655167360036780894579923693, −6.18119996349557034845715173650, −5.38943781518919203046266981553, −4.35635188458020468337971909209, −3.74131552264969916001740768214, −2.56100977990067252553594394880, −0.78258933533769630609459500687,
0.66370069951966065907765649761, 2.24603293672649670424589853110, 3.17512173195372653193629663407, 4.29392024536477669311303045219, 5.73648311835224548715439573185, 5.92255289785252634124047291822, 6.79580121814763522725822710827, 7.58391356396562637697442744786, 8.676683686322070143775019249456, 9.378468980958517742437818312667
