L(s) = 1 | + (2.37 + 2.37i)5-s + 3.64i·7-s + (−0.841 − 0.841i)11-s + (2.64 − 2.64i)13-s + 3.06·17-s + (1.64 − 1.64i)19-s + 7.82i·23-s + 6.29i·25-s + (0.692 − 0.692i)29-s + 0.354·31-s + (−8.66 + 8.66i)35-s + (−4.64 − 4.64i)37-s + 6.43i·41-s + (−5.64 − 5.64i)43-s − 11.1·47-s + ⋯ |
L(s) = 1 | + (1.06 + 1.06i)5-s + 1.37i·7-s + (−0.253 − 0.253i)11-s + (0.733 − 0.733i)13-s + 0.744·17-s + (0.377 − 0.377i)19-s + 1.63i·23-s + 1.25i·25-s + (0.128 − 0.128i)29-s + 0.0636·31-s + (−1.46 + 1.46i)35-s + (−0.763 − 0.763i)37-s + 1.00i·41-s + (−0.860 − 0.860i)43-s − 1.63·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.179 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.179 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.987466431\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.987466431\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.37 - 2.37i)T + 5iT^{2} \) |
| 7 | \( 1 - 3.64iT - 7T^{2} \) |
| 11 | \( 1 + (0.841 + 0.841i)T + 11iT^{2} \) |
| 13 | \( 1 + (-2.64 + 2.64i)T - 13iT^{2} \) |
| 17 | \( 1 - 3.06T + 17T^{2} \) |
| 19 | \( 1 + (-1.64 + 1.64i)T - 19iT^{2} \) |
| 23 | \( 1 - 7.82iT - 23T^{2} \) |
| 29 | \( 1 + (-0.692 + 0.692i)T - 29iT^{2} \) |
| 31 | \( 1 - 0.354T + 31T^{2} \) |
| 37 | \( 1 + (4.64 + 4.64i)T + 37iT^{2} \) |
| 41 | \( 1 - 6.43iT - 41T^{2} \) |
| 43 | \( 1 + (5.64 + 5.64i)T + 43iT^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 + (-5.44 - 5.44i)T + 53iT^{2} \) |
| 59 | \( 1 + (-7.82 - 7.82i)T + 59iT^{2} \) |
| 61 | \( 1 + (4.64 - 4.64i)T - 61iT^{2} \) |
| 67 | \( 1 + (-4 + 4i)T - 67iT^{2} \) |
| 71 | \( 1 + 3.36iT - 71T^{2} \) |
| 73 | \( 1 + 7.29iT - 73T^{2} \) |
| 79 | \( 1 + 4.35T + 79T^{2} \) |
| 83 | \( 1 + (-0.841 + 0.841i)T - 83iT^{2} \) |
| 89 | \( 1 - 9.50iT - 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−9.952820252178216666430525268621, −9.280357527278673364241437348241, −8.422687166895806527121041854682, −7.47816826028291669798731028744, −6.45239554157849800388009798006, −5.66821110320669532876719850247, −5.33527849621289582945499089204, −3.40917565127346452779587533444, −2.79109126198268768784270056265, −1.68223547966626059777731504254,
0.927120962253340371606931743162, 1.87694649151343520713365953277, 3.50169875901892915645325533870, 4.52641097576629326992136011686, 5.22135081753847673021489290128, 6.30864033587106380721248224539, 7.00505571594132257489644918191, 8.168155710568576188183617183196, 8.737524250270454690649294675272, 9.918582210049263310962775064758