| L(s) = 1 | − 0.691·2-s − 0.832·3-s − 1.52·4-s + 1.28·5-s + 0.574·6-s + 1.22·7-s + 2.43·8-s − 2.30·9-s − 0.888·10-s − 3.04·11-s + 1.26·12-s + 0.943·13-s − 0.848·14-s − 1.06·15-s + 1.36·16-s − 3.09·17-s + 1.59·18-s + 4.71·19-s − 1.95·20-s − 1.02·21-s + 2.10·22-s − 1.63·23-s − 2.02·24-s − 3.34·25-s − 0.652·26-s + 4.41·27-s − 1.86·28-s + ⋯ |
| L(s) = 1 | − 0.488·2-s − 0.480·3-s − 0.761·4-s + 0.574·5-s + 0.234·6-s + 0.464·7-s + 0.860·8-s − 0.769·9-s − 0.280·10-s − 0.917·11-s + 0.365·12-s + 0.261·13-s − 0.226·14-s − 0.276·15-s + 0.340·16-s − 0.750·17-s + 0.375·18-s + 1.08·19-s − 0.437·20-s − 0.222·21-s + 0.448·22-s − 0.341·23-s − 0.413·24-s − 0.669·25-s − 0.127·26-s + 0.849·27-s − 0.353·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 619 | \( 1 + T \) |
| good | 2 | \( 1 + 0.691T + 2T^{2} \) |
| 3 | \( 1 + 0.832T + 3T^{2} \) |
| 5 | \( 1 - 1.28T + 5T^{2} \) |
| 7 | \( 1 - 1.22T + 7T^{2} \) |
| 11 | \( 1 + 3.04T + 11T^{2} \) |
| 13 | \( 1 - 0.943T + 13T^{2} \) |
| 17 | \( 1 + 3.09T + 17T^{2} \) |
| 19 | \( 1 - 4.71T + 19T^{2} \) |
| 23 | \( 1 + 1.63T + 23T^{2} \) |
| 29 | \( 1 - 1.93T + 29T^{2} \) |
| 31 | \( 1 + 5.72T + 31T^{2} \) |
| 37 | \( 1 + 8.16T + 37T^{2} \) |
| 41 | \( 1 + 3.34T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 + 8.25T + 47T^{2} \) |
| 53 | \( 1 - 0.288T + 53T^{2} \) |
| 59 | \( 1 + 15.1T + 59T^{2} \) |
| 61 | \( 1 - 5.79T + 61T^{2} \) |
| 67 | \( 1 - 7.33T + 67T^{2} \) |
| 71 | \( 1 - 5.98T + 71T^{2} \) |
| 73 | \( 1 + 4.11T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 - 15.9T + 83T^{2} \) |
| 89 | \( 1 + 8.30T + 89T^{2} \) |
| 97 | \( 1 + 8.03T + 97T^{2} \) |
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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
−10.14417799786194447575749289960, −9.351376838170113609924231406682, −8.446748892640260195638775680421, −7.82487629524743006465891577606, −6.53342337658377603845173812020, −5.34179956570345318274301581285, −4.98292775613339626436984962789, −3.41444645015303315140352625242, −1.79115975531778778483527808172, 0,
1.79115975531778778483527808172, 3.41444645015303315140352625242, 4.98292775613339626436984962789, 5.34179956570345318274301581285, 6.53342337658377603845173812020, 7.82487629524743006465891577606, 8.446748892640260195638775680421, 9.351376838170113609924231406682, 10.14417799786194447575749289960
