L(s) = 1 | + 0.381i·3-s − 2.61i·5-s + 2.85·9-s + 3.46i·11-s − 2.14·13-s + 15-s + (−3.46 + 2.23i)17-s − 5.60·19-s − 5.60i·23-s − 1.85·25-s + 2.23i·27-s − 7.74i·29-s − 1.85i·31-s − 1.32·33-s − 1.32i·37-s + ⋯ |
L(s) = 1 | + 0.220i·3-s − 1.17i·5-s + 0.951·9-s + 1.04i·11-s − 0.593·13-s + 0.258·15-s + (−0.840 + 0.542i)17-s − 1.28·19-s − 1.16i·23-s − 0.370·25-s + 0.430i·27-s − 1.43i·29-s − 0.333i·31-s − 0.230·33-s − 0.217i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.840 + 0.542i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.840 + 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7613852725\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7613852725\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (3.46 - 2.23i)T \) |
good | 3 | \( 1 - 0.381iT - 3T^{2} \) |
| 5 | \( 1 + 2.61iT - 5T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 2.14T + 13T^{2} \) |
| 19 | \( 1 + 5.60T + 19T^{2} \) |
| 23 | \( 1 + 5.60iT - 23T^{2} \) |
| 29 | \( 1 + 7.74iT - 29T^{2} \) |
| 31 | \( 1 + 1.85iT - 31T^{2} \) |
| 37 | \( 1 + 1.32iT - 37T^{2} \) |
| 41 | \( 1 + 0.381iT - 41T^{2} \) |
| 43 | \( 1 + 0.854T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 + 1.85T + 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 + 8.56iT - 61T^{2} \) |
| 67 | \( 1 + 7.56T + 67T^{2} \) |
| 71 | \( 1 - 1.32iT - 71T^{2} \) |
| 73 | \( 1 + 10.8iT - 73T^{2} \) |
| 79 | \( 1 + 7.74iT - 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 7.85iT - 97T^{2} \) |
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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
−8.339189131075492524827099326526, −7.71114935806115214849002836008, −6.76956551243054078412338871980, −6.16101778698245373208729904671, −4.86817197436655579143017265528, −4.55620175884211994595985532122, −3.99321275277394649916316388102, −2.36802483371766203422864958819, −1.66745383720972612140427375616, −0.21991285521800350357138704208,
1.45399751469632690037250208671, 2.55088805469103063641341980893, 3.29822668015596840147180587136, 4.21666664647768923701789472020, 5.13972483001251989712241037319, 6.13915369708728576619654714346, 6.85965335443563396925250393904, 7.18557227142769825872816760763, 8.111269664040519195436702448473, 8.924512361925272452571974562552