L(s) = 1 | + (1 − i)5-s + 4i·7-s + (4 − 4i)11-s + (−3 − 3i)13-s − 6·17-s + (−4 − 4i)19-s + 8i·23-s + 3i·25-s + (3 + 3i)29-s − 4·31-s + (4 + 4i)35-s + (−1 + i)37-s + 2i·41-s + (−4 + 4i)43-s + 8·47-s + ⋯ |
L(s) = 1 | + (0.447 − 0.447i)5-s + 1.51i·7-s + (1.20 − 1.20i)11-s + (−0.832 − 0.832i)13-s − 1.45·17-s + (−0.917 − 0.917i)19-s + 1.66i·23-s + 0.600i·25-s + (0.557 + 0.557i)29-s − 0.718·31-s + (0.676 + 0.676i)35-s + (−0.164 + 0.164i)37-s + 0.312i·41-s + (−0.609 + 0.609i)43-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.110692162\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.110692162\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1 + i)T - 5iT^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + (-4 + 4i)T - 11iT^{2} \) |
| 13 | \( 1 + (3 + 3i)T + 13iT^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + (4 + 4i)T + 19iT^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + (-3 - 3i)T + 29iT^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (1 - i)T - 37iT^{2} \) |
| 41 | \( 1 - 2iT - 41T^{2} \) |
| 43 | \( 1 + (4 - 4i)T - 43iT^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + (7 - 7i)T - 53iT^{2} \) |
| 59 | \( 1 - 59iT^{2} \) |
| 61 | \( 1 + (-3 - 3i)T + 61iT^{2} \) |
| 67 | \( 1 + (-8 - 8i)T + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + (4 + 4i)T + 83iT^{2} \) |
| 89 | \( 1 - 16iT - 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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
−8.761434945972377780073290522973, −8.036087874543968657123952715208, −6.91484434454750907714540599404, −6.30926654015379176875511376998, −5.50431227414308263719522758654, −5.13963720823613379688965916450, −4.03846636892798658589323392156, −3.01527363462471628898869019071, −2.29528866876146208117454807243, −1.25165486099068251958118881924,
0.29341512576435437368663662385, 1.83828472324304692228693927546, 2.28260836630970977932833247742, 3.82849251732180625923887373814, 4.30529412858949967142457514204, 4.78734963470395720291513276016, 6.38641468556251992990527142020, 6.64148286105137298794016338169, 7.09220843362054680323871305196, 8.012840779705286885830384815674