L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (0.707 + 0.707i)5-s + (0.866 + 0.5i)7-s + (−0.707 − 0.707i)8-s + (−0.500 − 0.866i)10-s + (−0.448 − 0.258i)11-s + (−0.707 − 0.707i)14-s + (0.500 + 0.866i)16-s + (0.258 + 0.965i)20-s + (0.366 + 0.366i)22-s + 1.00i·25-s + (0.5 + 0.866i)28-s + 1.93·29-s + (0.866 − 1.5i)31-s + (−0.258 − 0.965i)32-s + ⋯ |
L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (0.707 + 0.707i)5-s + (0.866 + 0.5i)7-s + (−0.707 − 0.707i)8-s + (−0.500 − 0.866i)10-s + (−0.448 − 0.258i)11-s + (−0.707 − 0.707i)14-s + (0.500 + 0.866i)16-s + (0.258 + 0.965i)20-s + (0.366 + 0.366i)22-s + 1.00i·25-s + (0.5 + 0.866i)28-s + 1.93·29-s + (0.866 − 1.5i)31-s + (−0.258 − 0.965i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9917626900\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9917626900\) |
\(L(1)\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
good | 11 | \( 1 + (0.448 + 0.258i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 - 1.93T + T^{2} \) |
| 31 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (1.67 - 0.448i)T + (0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.366 + 0.366i)T + iT^{2} \) |
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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
−9.246331416618042261693226018259, −8.293294346443927487894262259859, −7.935277524082281504684187008714, −6.92841494369340286222342774915, −6.22201270614297051236477309924, −5.47955666897045606391904109961, −4.32759430801120907449059186093, −2.92736773871478313915665158304, −2.44095899126128940688148227642, −1.33321435150737627234104935695,
1.05039777628686650185133693215, 1.90674005539211775509127834866, 2.99996750560294479090601568455, 4.67832418955061089509713006144, 5.04411938265067371551322309868, 6.13643792120203269063979196918, 6.79584671537380959230109812991, 7.75689626815562146118682246519, 8.331335763043498190253460655864, 8.888827243907044449168299534678