L(s) = 1 | + (−0.258 + 0.965i)2-s + (−0.707 + 0.707i)3-s + (−0.866 − 0.499i)4-s + (−0.500 − 0.866i)6-s + (0.707 − 0.707i)8-s − 1.00i·9-s + (0.965 − 0.258i)12-s + (0.500 + 0.866i)16-s + (1.41 + 1.41i)17-s + (0.965 + 0.258i)18-s − i·19-s + i·24-s + (0.707 + 0.707i)27-s + (−0.965 + 0.258i)32-s + (−1.73 + 1.00i)34-s + ⋯ |
L(s) = 1 | + (−0.258 + 0.965i)2-s + (−0.707 + 0.707i)3-s + (−0.866 − 0.499i)4-s + (−0.500 − 0.866i)6-s + (0.707 − 0.707i)8-s − 1.00i·9-s + (0.965 − 0.258i)12-s + (0.500 + 0.866i)16-s + (1.41 + 1.41i)17-s + (0.965 + 0.258i)18-s − i·19-s + i·24-s + (0.707 + 0.707i)27-s + (−0.965 + 0.258i)32-s + (−1.73 + 1.00i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7145169227\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7145169227\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 - 0.965i)T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 19 | \( 1 + iT - T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.448 + 1.67i)T + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 - 1.73T + T^{2} \) |
| 97 | \( 1 + (1.67 + 0.448i)T + (0.866 + 0.5i)T^{2} \) |
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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
−9.561389540031988088865156655943, −9.059709104164981330523856125334, −8.063400574693650083019136006483, −7.38658554919689517101239304191, −6.22415234123759863791052863632, −5.96186106408533983196218075924, −4.92759265151079259715679660242, −4.27342018031380039672694363437, −3.24329077490562343282740099791, −1.16423244514000774639841882485,
0.815501324561288130351982581728, 1.97360142718853277560915551834, 3.02955244935829610636045908990, 4.11353697261956627823289694751, 5.24055957727984368598175476072, 5.72629474717964748050886325106, 7.08177321246319718839540973977, 7.63584384205201546866811586109, 8.419627107563758700468507482547, 9.392513656807234442262780525609