| L(s) = 1 | − 2.50·2-s + 0.797·3-s + 4.29·4-s − 3.82·5-s − 2.00·6-s − 2.73·7-s − 5.74·8-s − 2.36·9-s + 9.59·10-s − 5.60·11-s + 3.42·12-s − 3.88·13-s + 6.87·14-s − 3.05·15-s + 5.83·16-s + 5.63·17-s + 5.92·18-s + 2.10·19-s − 16.4·20-s − 2.18·21-s + 14.0·22-s + 7.90·23-s − 4.58·24-s + 9.64·25-s + 9.73·26-s − 4.27·27-s − 11.7·28-s + ⋯ |
| L(s) = 1 | − 1.77·2-s + 0.460·3-s + 2.14·4-s − 1.71·5-s − 0.816·6-s − 1.03·7-s − 2.03·8-s − 0.787·9-s + 3.03·10-s − 1.69·11-s + 0.988·12-s − 1.07·13-s + 1.83·14-s − 0.788·15-s + 1.45·16-s + 1.36·17-s + 1.39·18-s + 0.482·19-s − 3.67·20-s − 0.476·21-s + 2.99·22-s + 1.64·23-s − 0.936·24-s + 1.92·25-s + 1.90·26-s − 0.823·27-s − 2.22·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 6047 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.50T + 2T^{2} \) |
| 3 | \( 1 - 0.797T + 3T^{2} \) |
| 5 | \( 1 + 3.82T + 5T^{2} \) |
| 7 | \( 1 + 2.73T + 7T^{2} \) |
| 11 | \( 1 + 5.60T + 11T^{2} \) |
| 13 | \( 1 + 3.88T + 13T^{2} \) |
| 17 | \( 1 - 5.63T + 17T^{2} \) |
| 19 | \( 1 - 2.10T + 19T^{2} \) |
| 23 | \( 1 - 7.90T + 23T^{2} \) |
| 29 | \( 1 - 0.651T + 29T^{2} \) |
| 31 | \( 1 + 5.09T + 31T^{2} \) |
| 37 | \( 1 + 6.71T + 37T^{2} \) |
| 41 | \( 1 + 4.35T + 41T^{2} \) |
| 43 | \( 1 + 1.70T + 43T^{2} \) |
| 47 | \( 1 - 4.66T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 - 9.47T + 59T^{2} \) |
| 61 | \( 1 - 2.91T + 61T^{2} \) |
| 67 | \( 1 + 1.65T + 67T^{2} \) |
| 71 | \( 1 - 0.503T + 71T^{2} \) |
| 73 | \( 1 + 9.84T + 73T^{2} \) |
| 79 | \( 1 + 3.32T + 79T^{2} \) |
| 83 | \( 1 - 1.28T + 83T^{2} \) |
| 89 | \( 1 - 5.89T + 89T^{2} \) |
| 97 | \( 1 - 6.57T + 97T^{2} \) |
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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
−7.78243816468044085773944834212, −7.25558133758488229082397813438, −7.07966839140454255156595116600, −5.66084327621641276103563994798, −4.97521352537763251125068936328, −3.42036177236447502267493632129, −3.11638912099366410450875764941, −2.37451396063610258596398620250, −0.70214303803936454212013172738, 0,
0.70214303803936454212013172738, 2.37451396063610258596398620250, 3.11638912099366410450875764941, 3.42036177236447502267493632129, 4.97521352537763251125068936328, 5.66084327621641276103563994798, 7.07966839140454255156595116600, 7.25558133758488229082397813438, 7.78243816468044085773944834212
