| L(s) = 1 | − 2.77·2-s + 3-s + 5.71·4-s + 1.71·5-s − 2.77·6-s − 2.08·7-s − 10.3·8-s + 9-s − 4.77·10-s − 0.368·11-s + 5.71·12-s − 1.86·13-s + 5.78·14-s + 1.71·15-s + 17.2·16-s + 17-s − 2.77·18-s − 4.42·19-s + 9.82·20-s − 2.08·21-s + 1.02·22-s + 0.403·23-s − 10.3·24-s − 2.05·25-s + 5.17·26-s + 27-s − 11.9·28-s + ⋯ |
| L(s) = 1 | − 1.96·2-s + 0.577·3-s + 2.85·4-s + 0.768·5-s − 1.13·6-s − 0.787·7-s − 3.65·8-s + 0.333·9-s − 1.50·10-s − 0.110·11-s + 1.65·12-s − 0.516·13-s + 1.54·14-s + 0.443·15-s + 4.31·16-s + 0.242·17-s − 0.654·18-s − 1.01·19-s + 2.19·20-s − 0.454·21-s + 0.218·22-s + 0.0840·23-s − 2.10·24-s − 0.410·25-s + 1.01·26-s + 0.192·27-s − 2.25·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 + 2.77T + 2T^{2} \) |
| 5 | \( 1 - 1.71T + 5T^{2} \) |
| 7 | \( 1 + 2.08T + 7T^{2} \) |
| 11 | \( 1 + 0.368T + 11T^{2} \) |
| 13 | \( 1 + 1.86T + 13T^{2} \) |
| 19 | \( 1 + 4.42T + 19T^{2} \) |
| 23 | \( 1 - 0.403T + 23T^{2} \) |
| 29 | \( 1 + 7.21T + 29T^{2} \) |
| 31 | \( 1 - 6.40T + 31T^{2} \) |
| 37 | \( 1 - 8.74T + 37T^{2} \) |
| 41 | \( 1 + 3.56T + 41T^{2} \) |
| 43 | \( 1 - 6.08T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + 9.41T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 83 | \( 1 - 7.66T + 83T^{2} \) |
| 89 | \( 1 - 0.676T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−8.078519671155550129197706933466, −7.71258603417635450016533983868, −6.75766936277411572417102402985, −6.30993997362583781989076200970, −5.51549474253871004061814486434, −3.91993111708529199586682165340, −2.74762394977558262288265688362, −2.33520077690526467620605503134, −1.31848414625060974512293590777, 0,
1.31848414625060974512293590777, 2.33520077690526467620605503134, 2.74762394977558262288265688362, 3.91993111708529199586682165340, 5.51549474253871004061814486434, 6.30993997362583781989076200970, 6.75766936277411572417102402985, 7.71258603417635450016533983868, 8.078519671155550129197706933466
