| L(s) = 1 | + 2.14·2-s + 3-s + 2.61·4-s + 0.492·5-s + 2.14·6-s − 0.251·7-s + 1.32·8-s + 9-s + 1.05·10-s − 3.24·11-s + 2.61·12-s + 4.64·13-s − 0.539·14-s + 0.492·15-s − 2.38·16-s − 3.90·17-s + 2.14·18-s − 1.90·19-s + 1.28·20-s − 0.251·21-s − 6.97·22-s + 8.28·23-s + 1.32·24-s − 4.75·25-s + 9.99·26-s + 27-s − 0.657·28-s + ⋯ |
| L(s) = 1 | + 1.51·2-s + 0.577·3-s + 1.30·4-s + 0.220·5-s + 0.877·6-s − 0.0949·7-s + 0.469·8-s + 0.333·9-s + 0.334·10-s − 0.978·11-s + 0.755·12-s + 1.28·13-s − 0.144·14-s + 0.127·15-s − 0.595·16-s − 0.947·17-s + 0.506·18-s − 0.437·19-s + 0.288·20-s − 0.0548·21-s − 1.48·22-s + 1.72·23-s + 0.271·24-s − 0.951·25-s + 1.95·26-s + 0.192·27-s − 0.124·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.374520288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.374520288\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 2.14T + 2T^{2} \) |
| 5 | \( 1 - 0.492T + 5T^{2} \) |
| 7 | \( 1 + 0.251T + 7T^{2} \) |
| 11 | \( 1 + 3.24T + 11T^{2} \) |
| 13 | \( 1 - 4.64T + 13T^{2} \) |
| 17 | \( 1 + 3.90T + 17T^{2} \) |
| 19 | \( 1 + 1.90T + 19T^{2} \) |
| 23 | \( 1 - 8.28T + 23T^{2} \) |
| 29 | \( 1 + 9.34T + 29T^{2} \) |
| 31 | \( 1 + 0.920T + 31T^{2} \) |
| 37 | \( 1 - 8.55T + 37T^{2} \) |
| 41 | \( 1 + 2.98T + 41T^{2} \) |
| 43 | \( 1 - 1.41T + 43T^{2} \) |
| 47 | \( 1 - 7.43T + 47T^{2} \) |
| 53 | \( 1 + 1.05T + 53T^{2} \) |
| 59 | \( 1 - 6.89T + 59T^{2} \) |
| 61 | \( 1 + 5.54T + 61T^{2} \) |
| 67 | \( 1 - 9.13T + 67T^{2} \) |
| 71 | \( 1 + 4.84T + 71T^{2} \) |
| 73 | \( 1 - 8.54T + 73T^{2} \) |
| 79 | \( 1 - 2.23T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + 3.28T + 89T^{2} \) |
| 97 | \( 1 - 4.95T + 97T^{2} \) |
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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
−11.33931077486783160899096307794, −10.88743115050040352766973347156, −9.457778550213393612526371250126, −8.571483139430226575536850318508, −7.35643992949541726598146729718, −6.29083556668186871853295422833, −5.40803480449463684867413455758, −4.29470198996041733446677891896, −3.32946799372827516103099625928, −2.20638230963401454952380653548,
2.20638230963401454952380653548, 3.32946799372827516103099625928, 4.29470198996041733446677891896, 5.40803480449463684867413455758, 6.29083556668186871853295422833, 7.35643992949541726598146729718, 8.571483139430226575536850318508, 9.457778550213393612526371250126, 10.88743115050040352766973347156, 11.33931077486783160899096307794
