L(s) = 1 | + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.707 − 1.22i)5-s − i·7-s + (0.707 + 0.707i)8-s + (−0.366 − 1.36i)10-s + 1.41i·11-s + (0.5 − 0.866i)13-s + (0.258 − 0.965i)14-s + (0.500 + 0.866i)16-s + i·19-s − 1.41i·20-s + (−0.366 + 1.36i)22-s + (−1.22 − 0.707i)23-s + (−0.499 + 0.866i)25-s + (0.707 − 0.707i)26-s + ⋯ |
L(s) = 1 | + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.707 − 1.22i)5-s − i·7-s + (0.707 + 0.707i)8-s + (−0.366 − 1.36i)10-s + 1.41i·11-s + (0.5 − 0.866i)13-s + (0.258 − 0.965i)14-s + (0.500 + 0.866i)16-s + i·19-s − 1.41i·20-s + (−0.366 + 1.36i)22-s + (−1.22 − 0.707i)23-s + (−0.499 + 0.866i)25-s + (0.707 − 0.707i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.502031303\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.502031303\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 - 0.258i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - iT \) |
good | 5 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + iT - T^{2} \) |
| 11 | \( 1 - 1.41iT - T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - iT - T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
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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
−10.65289158505931987992786909146, −10.07615495091473951242855859389, −8.597908888672023038064110548448, −7.87673444378264271477841716317, −7.22645662142515983275379613371, −6.07691279855582409247832469859, −4.91371105985586427181296576014, −4.33196971939661838240680503940, −3.48221391099668650963820352459, −1.62463616756399609463386000202,
2.20776088424371355918668987536, 3.24101108736264869729566913663, 3.93892080431188135125227837304, 5.37185667634884437118104599082, 6.21727926806713274991732146427, 6.88730515362269825195815864517, 7.985395863959707180681766163992, 8.999310516052095031435843537619, 10.17446997074512643735909550855, 11.19237866207424476557748299959