L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (−1.46 − 1.12i)5-s + (−0.707 + 0.707i)8-s + (0.965 + 0.258i)9-s + (−1.46 + 1.12i)10-s + 2i·13-s + (0.500 + 0.866i)16-s + (0.258 + 0.965i)17-s + (0.499 − 0.866i)18-s + (0.707 + 1.70i)20-s + (0.624 + 2.33i)25-s + (1.93 + 0.517i)26-s + (−0.707 + 0.292i)29-s + (0.965 − 0.258i)32-s + ⋯ |
L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (−1.46 − 1.12i)5-s + (−0.707 + 0.707i)8-s + (0.965 + 0.258i)9-s + (−1.46 + 1.12i)10-s + 2i·13-s + (0.500 + 0.866i)16-s + (0.258 + 0.965i)17-s + (0.499 − 0.866i)18-s + (0.707 + 1.70i)20-s + (0.624 + 2.33i)25-s + (1.93 + 0.517i)26-s + (−0.707 + 0.292i)29-s + (0.965 − 0.258i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8191737631\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8191737631\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 + 0.965i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (-0.258 - 0.965i)T \) |
good | 3 | \( 1 + (-0.965 - 0.258i)T^{2} \) |
| 5 | \( 1 + (1.46 + 1.12i)T + (0.258 + 0.965i)T^{2} \) |
| 11 | \( 1 + (0.258 - 0.965i)T^{2} \) |
| 13 | \( 1 - 2iT - T^{2} \) |
| 19 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (-0.965 + 0.258i)T^{2} \) |
| 29 | \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 + (-0.965 - 0.258i)T^{2} \) |
| 37 | \( 1 + (0.465 - 0.607i)T + (-0.258 - 0.965i)T^{2} \) |
| 41 | \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (0.0999 + 0.758i)T + (-0.965 + 0.258i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 73 | \( 1 + (0.241 - 1.83i)T + (-0.965 - 0.258i)T^{2} \) |
| 79 | \( 1 + (-0.965 + 0.258i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \) |
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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
−8.828787554331482704421865163711, −8.283806123655658334670551238666, −7.40852773795306905196952405955, −6.60252488161988554987392414740, −5.30888868823132809437767433628, −4.63677126789104023780023226718, −4.01169757331518484525035510511, −3.63811592093273672741225142519, −1.97444567019157911734429425026, −1.24184580967821374918365488716,
0.52099088883558125701617172638, 2.86033546862184859570663045655, 3.46595150318003168243900735760, 4.14768212297803959342942287360, 5.06213924755462749641537326424, 5.91807914916219672274580449215, 6.87437439816688663166071003731, 7.38268317642748898704292462156, 7.77009642318559194963026993765, 8.460821619069396824450004124696