| L(s) = 1 | − 1.84·2-s + 2.72·3-s + 1.41·4-s + 1.47·5-s − 5.04·6-s − 2.80·7-s + 1.08·8-s + 4.44·9-s − 2.71·10-s + 11-s + 3.85·12-s − 6.14·13-s + 5.18·14-s + 4.01·15-s − 4.82·16-s − 4.41·17-s − 8.21·18-s − 7.60·19-s + 2.07·20-s − 7.66·21-s − 1.84·22-s − 0.405·23-s + 2.95·24-s − 2.83·25-s + 11.3·26-s + 3.94·27-s − 3.96·28-s + ⋯ |
| L(s) = 1 | − 1.30·2-s + 1.57·3-s + 0.706·4-s + 0.657·5-s − 2.05·6-s − 1.06·7-s + 0.383·8-s + 1.48·9-s − 0.858·10-s + 0.301·11-s + 1.11·12-s − 1.70·13-s + 1.38·14-s + 1.03·15-s − 1.20·16-s − 1.07·17-s − 1.93·18-s − 1.74·19-s + 0.464·20-s − 1.67·21-s − 0.393·22-s − 0.0844·23-s + 0.603·24-s − 0.567·25-s + 2.22·26-s + 0.758·27-s − 0.749·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 2 | \( 1 + 1.84T + 2T^{2} \) |
| 3 | \( 1 - 2.72T + 3T^{2} \) |
| 5 | \( 1 - 1.47T + 5T^{2} \) |
| 7 | \( 1 + 2.80T + 7T^{2} \) |
| 13 | \( 1 + 6.14T + 13T^{2} \) |
| 17 | \( 1 + 4.41T + 17T^{2} \) |
| 19 | \( 1 + 7.60T + 19T^{2} \) |
| 23 | \( 1 + 0.405T + 23T^{2} \) |
| 29 | \( 1 + 3.69T + 29T^{2} \) |
| 31 | \( 1 - 1.28T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 - 5.02T + 41T^{2} \) |
| 43 | \( 1 - 4.21T + 43T^{2} \) |
| 47 | \( 1 - 8.45T + 47T^{2} \) |
| 53 | \( 1 + 5.29T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 1.76T + 61T^{2} \) |
| 67 | \( 1 - 9.56T + 67T^{2} \) |
| 71 | \( 1 + 0.0487T + 71T^{2} \) |
| 73 | \( 1 - 16.1T + 73T^{2} \) |
| 79 | \( 1 - 3.58T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 0.445T + 89T^{2} \) |
| 97 | \( 1 + 3.24T + 97T^{2} \) |
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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
−9.201425492141819868677526077771, −8.624669940865888260772643331205, −7.73479789494271002515813618729, −7.05164849059003353326487171951, −6.29505288067427780074380079309, −4.69420981895452134336817531198, −3.71084540970582230185077931260, −2.32865795881624830155858192142, −2.09109163832579788551920021833, 0,
2.09109163832579788551920021833, 2.32865795881624830155858192142, 3.71084540970582230185077931260, 4.69420981895452134336817531198, 6.29505288067427780074380079309, 7.05164849059003353326487171951, 7.73479789494271002515813618729, 8.624669940865888260772643331205, 9.201425492141819868677526077771
