L(s) = 1 | − 1.71·2-s − 2.73·3-s + 0.935·4-s − 2.57·5-s + 4.67·6-s + 0.316·7-s + 1.82·8-s + 4.45·9-s + 4.41·10-s + 4.24·11-s − 2.55·12-s − 13-s − 0.542·14-s + 7.04·15-s − 4.99·16-s − 1.03·17-s − 7.63·18-s − 0.275·19-s − 2.41·20-s − 0.864·21-s − 7.28·22-s − 6.05·23-s − 4.97·24-s + 1.65·25-s + 1.71·26-s − 3.96·27-s + 0.296·28-s + ⋯ |
L(s) = 1 | − 1.21·2-s − 1.57·3-s + 0.467·4-s − 1.15·5-s + 1.90·6-s + 0.119·7-s + 0.644·8-s + 1.48·9-s + 1.39·10-s + 1.28·11-s − 0.737·12-s − 0.277·13-s − 0.145·14-s + 1.81·15-s − 1.24·16-s − 0.250·17-s − 1.79·18-s − 0.0631·19-s − 0.539·20-s − 0.188·21-s − 1.55·22-s − 1.26·23-s − 1.01·24-s + 0.330·25-s + 0.336·26-s − 0.763·27-s + 0.0559·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2258623599\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2258623599\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 + 1.71T + 2T^{2} \) |
| 3 | \( 1 + 2.73T + 3T^{2} \) |
| 5 | \( 1 + 2.57T + 5T^{2} \) |
| 7 | \( 1 - 0.316T + 7T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 17 | \( 1 + 1.03T + 17T^{2} \) |
| 19 | \( 1 + 0.275T + 19T^{2} \) |
| 23 | \( 1 + 6.05T + 23T^{2} \) |
| 29 | \( 1 + 1.93T + 29T^{2} \) |
| 31 | \( 1 - 2.97T + 31T^{2} \) |
| 37 | \( 1 + 6.71T + 37T^{2} \) |
| 41 | \( 1 + 0.795T + 41T^{2} \) |
| 43 | \( 1 + 12.9T + 43T^{2} \) |
| 47 | \( 1 - 5.23T + 47T^{2} \) |
| 53 | \( 1 - 7.01T + 53T^{2} \) |
| 59 | \( 1 - 8.54T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 - 7.34T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 5.63T + 73T^{2} \) |
| 79 | \( 1 + 6.11T + 79T^{2} \) |
| 83 | \( 1 + 4.80T + 83T^{2} \) |
| 89 | \( 1 + 3.00T + 89T^{2} \) |
| 97 | \( 1 + 4.70T + 97T^{2} \) |
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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
−9.837436693314586470847621658125, −8.715082716187376940659821156386, −8.118012697349669199805997631619, −7.07748555577696129430111669895, −6.68432631271483466612264857637, −5.51456683997437598213276022916, −4.48895787051104244807895637342, −3.85170311041726921010159527121, −1.66877347307079328758562084761, −0.46342125418261512495764289402,
0.46342125418261512495764289402, 1.66877347307079328758562084761, 3.85170311041726921010159527121, 4.48895787051104244807895637342, 5.51456683997437598213276022916, 6.68432631271483466612264857637, 7.07748555577696129430111669895, 8.118012697349669199805997631619, 8.715082716187376940659821156386, 9.837436693314586470847621658125