L(s) = 1 | + (0.415 − 0.909i)2-s + (−1.10 + 1.27i)3-s + (−0.654 − 0.755i)4-s + (0.698 + 1.53i)6-s + (−0.959 + 0.281i)8-s + (−0.260 − 1.81i)9-s + (1.84 + 0.540i)11-s + 1.68·12-s + (−0.142 + 0.989i)16-s + (0.345 + 0.755i)17-s + (−1.75 − 0.515i)18-s + (0.273 + 1.89i)19-s + (1.25 − 1.45i)22-s + (0.698 − 1.53i)24-s + (0.841 − 0.540i)25-s + ⋯ |
L(s) = 1 | + (0.415 − 0.909i)2-s + (−1.10 + 1.27i)3-s + (−0.654 − 0.755i)4-s + (0.698 + 1.53i)6-s + (−0.959 + 0.281i)8-s + (−0.260 − 1.81i)9-s + (1.84 + 0.540i)11-s + 1.68·12-s + (−0.142 + 0.989i)16-s + (0.345 + 0.755i)17-s + (−1.75 − 0.515i)18-s + (0.273 + 1.89i)19-s + (1.25 − 1.45i)22-s + (0.698 − 1.53i)24-s + (0.841 − 0.540i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8087213915\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8087213915\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.415 + 0.909i)T \) |
| 89 | \( 1 + (-0.841 - 0.540i)T \) |
good | 3 | \( 1 + (1.10 - 1.27i)T + (-0.142 - 0.989i)T^{2} \) |
| 5 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 7 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 11 | \( 1 + (-1.84 - 0.540i)T + (0.841 + 0.540i)T^{2} \) |
| 13 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 17 | \( 1 + (-0.345 - 0.755i)T + (-0.654 + 0.755i)T^{2} \) |
| 19 | \( 1 + (-0.273 - 1.89i)T + (-0.959 + 0.281i)T^{2} \) |
| 23 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 29 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 31 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (1.30 + 1.51i)T + (-0.142 + 0.989i)T^{2} \) |
| 43 | \( 1 + (0.797 + 0.234i)T + (0.841 + 0.540i)T^{2} \) |
| 47 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 53 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 59 | \( 1 + (0.544 + 0.627i)T + (-0.142 + 0.989i)T^{2} \) |
| 61 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 67 | \( 1 + (-0.186 + 0.215i)T + (-0.142 - 0.989i)T^{2} \) |
| 71 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 73 | \( 1 + (0.239 - 1.66i)T + (-0.959 - 0.281i)T^{2} \) |
| 79 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 83 | \( 1 + (0.544 + 1.19i)T + (-0.654 + 0.755i)T^{2} \) |
| 97 | \( 1 + (-0.273 + 0.0801i)T + (0.841 - 0.540i)T^{2} \) |
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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
−10.49685784458442842337488737779, −10.16763733186725701567979825497, −9.385892421712434965113084258628, −8.536395538938519556273811742952, −6.66917924313449610972633469244, −5.89315110511703219679009049763, −5.06037876520149224652561533268, −3.99004821180041182528631374807, −3.66459011618508755048608459373, −1.53299712250101336412350124626,
1.06949721861341796834878220543, 3.09418354843629659759596006517, 4.60970154857928719592728625317, 5.43002126751151767725305623390, 6.51092618719336287092938970249, 6.75669480780390164991816341901, 7.57798652109391957475075850114, 8.727046719439326502479411222935, 9.430307114991187556488583493888, 11.07631637880200148114978022898