| L(s) = 1 | − 0.0714·2-s − 8.95·3-s − 7.99·4-s + 13.8·5-s + 0.640·6-s + 19.5·7-s + 1.14·8-s + 53.2·9-s − 0.991·10-s − 61.7·11-s + 71.6·12-s − 24.5·13-s − 1.39·14-s − 124.·15-s + 63.8·16-s − 12.3·17-s − 3.80·18-s − 35.5·19-s − 110.·20-s − 175.·21-s + 4.41·22-s − 22.2·23-s − 10.2·24-s + 67.2·25-s + 1.75·26-s − 234.·27-s − 156.·28-s + ⋯ |
| L(s) = 1 | − 0.0252·2-s − 1.72·3-s − 0.999·4-s + 1.24·5-s + 0.0435·6-s + 1.05·7-s + 0.0505·8-s + 1.97·9-s − 0.0313·10-s − 1.69·11-s + 1.72·12-s − 0.524·13-s − 0.0267·14-s − 2.13·15-s + 0.998·16-s − 0.175·17-s − 0.0498·18-s − 0.429·19-s − 1.23·20-s − 1.82·21-s + 0.0427·22-s − 0.201·23-s − 0.0871·24-s + 0.537·25-s + 0.0132·26-s − 1.67·27-s − 1.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( 1 + 0.0714T + 8T^{2} \) |
| 3 | \( 1 + 8.95T + 27T^{2} \) |
| 5 | \( 1 - 13.8T + 125T^{2} \) |
| 7 | \( 1 - 19.5T + 343T^{2} \) |
| 11 | \( 1 + 61.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 24.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 12.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 35.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 22.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 230.T + 2.43e4T^{2} \) |
| 37 | \( 1 - 196.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 9.13T + 6.89e4T^{2} \) |
| 43 | \( 1 - 450.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 115.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 252.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 413.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 493.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 611.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 392.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 358.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 274.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 723.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 702.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 969.T + 9.12e5T^{2} \) |
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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
−9.547444147896943110942279469751, −8.344194074312037775737615486846, −7.53732518598332300984033789043, −6.26264766132659538439599194012, −5.58009584730506299339352376999, −4.96219732952841289130761052326, −4.49812992174887101253543120885, −2.40739833051450751391783010658, −1.12773665455817615214291547063, 0,
1.12773665455817615214291547063, 2.40739833051450751391783010658, 4.49812992174887101253543120885, 4.96219732952841289130761052326, 5.58009584730506299339352376999, 6.26264766132659538439599194012, 7.53732518598332300984033789043, 8.344194074312037775737615486846, 9.547444147896943110942279469751
