| L(s) = 1 | − 2.46·2-s + 3.03·3-s + 4.06·4-s + 0.527·5-s − 7.46·6-s − 5.09·8-s + 6.19·9-s − 1.29·10-s + 5.19·11-s + 12.3·12-s − 0.508·13-s + 1.59·15-s + 4.41·16-s − 2.47·17-s − 15.2·18-s − 19-s + 2.14·20-s − 12.7·22-s + 2.60·23-s − 15.4·24-s − 4.72·25-s + 1.25·26-s + 9.68·27-s + 7.14·29-s − 3.94·30-s + 8.03·31-s − 0.685·32-s + ⋯ |
| L(s) = 1 | − 1.74·2-s + 1.75·3-s + 2.03·4-s + 0.235·5-s − 3.04·6-s − 1.80·8-s + 2.06·9-s − 0.411·10-s + 1.56·11-s + 3.56·12-s − 0.141·13-s + 0.413·15-s + 1.10·16-s − 0.599·17-s − 3.59·18-s − 0.229·19-s + 0.479·20-s − 2.72·22-s + 0.542·23-s − 3.15·24-s − 0.944·25-s + 0.245·26-s + 1.86·27-s + 1.32·29-s − 0.719·30-s + 1.44·31-s − 0.121·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.499831471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.499831471\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 2.46T + 2T^{2} \) |
| 3 | \( 1 - 3.03T + 3T^{2} \) |
| 5 | \( 1 - 0.527T + 5T^{2} \) |
| 11 | \( 1 - 5.19T + 11T^{2} \) |
| 13 | \( 1 + 0.508T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 23 | \( 1 - 2.60T + 23T^{2} \) |
| 29 | \( 1 - 7.14T + 29T^{2} \) |
| 31 | \( 1 - 8.03T + 31T^{2} \) |
| 37 | \( 1 + 4.76T + 37T^{2} \) |
| 41 | \( 1 + 4.92T + 41T^{2} \) |
| 43 | \( 1 + 8.66T + 43T^{2} \) |
| 47 | \( 1 - 1.54T + 47T^{2} \) |
| 53 | \( 1 - 0.0990T + 53T^{2} \) |
| 59 | \( 1 - 5.64T + 59T^{2} \) |
| 61 | \( 1 + 7.39T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + 0.577T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 - 4.97T + 79T^{2} \) |
| 83 | \( 1 + 0.753T + 83T^{2} \) |
| 89 | \( 1 - 2.73T + 89T^{2} \) |
| 97 | \( 1 - 0.0779T + 97T^{2} \) |
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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
−9.828994310691904322824476695124, −9.023280348809314490813425004057, −8.620036404660206121236848086005, −7.987895891694283402898219865783, −6.95222576872411367955775404614, −6.50150947293295405126898233198, −4.41585267168923823952323515345, −3.24203944256221060455529588386, −2.19975378651990435209091830791, −1.31810651374183008416775367719,
1.31810651374183008416775367719, 2.19975378651990435209091830791, 3.24203944256221060455529588386, 4.41585267168923823952323515345, 6.50150947293295405126898233198, 6.95222576872411367955775404614, 7.987895891694283402898219865783, 8.620036404660206121236848086005, 9.023280348809314490813425004057, 9.828994310691904322824476695124
