L(s) = 1 | − 10.3i·5-s − 14.6·7-s + 5.65i·11-s + 17·25-s − 218. i·29-s − 338.·31-s + 152. i·35-s − 127·49-s + 509. i·53-s + 58.7·55-s + 554. i·59-s + 322·73-s − 83.1i·77-s + 308.·79-s + 1.22e3i·83-s + ⋯ |
L(s) = 1 | − 0.929i·5-s − 0.793·7-s + 0.155i·11-s + 0.136·25-s − 1.39i·29-s − 1.95·31-s + 0.737i·35-s − 0.370·49-s + 1.31i·53-s + 0.144·55-s + 1.22i·59-s + 0.516·73-s − 0.123i·77-s + 0.439·79-s + 1.62i·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.6864387752\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6864387752\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + 10.3iT - 125T^{2} \) |
| 7 | \( 1 + 14.6T + 343T^{2} \) |
| 11 | \( 1 - 5.65iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 2.19e3T^{2} \) |
| 17 | \( 1 + 4.91e3T^{2} \) |
| 19 | \( 1 - 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 + 218. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 338.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 5.06e4T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 - 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 - 509. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 554. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 2.26e5T^{2} \) |
| 67 | \( 1 - 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 - 322T + 3.89e5T^{2} \) |
| 79 | \( 1 - 308.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.22e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 + 574T + 9.12e5T^{2} \) |
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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
−9.441693097634920580878082873735, −8.994071311433229844906527274484, −8.024461881209783311379158359721, −7.18188474678392013337117004315, −6.19473252602108113973153109669, −5.39774657732659590211850382896, −4.43200511806724335986078499005, −3.52243760349827180664154857187, −2.28398610328640160690189559074, −0.982462127589477152619674389567,
0.18634918642795975289557131741, 1.84192360072070073976371195044, 3.08922685911511800726255743696, 3.61358475823917058532946860367, 4.99262636771099118916621861472, 5.96649615572181673186670565505, 6.81474034523309764541307555146, 7.31278632568474624046090893972, 8.458493780205957967859883984360, 9.304285163120311695607075791229