L(s) = 1 | + (0.866 + 0.5i)3-s + (−0.866 − 0.5i)4-s + (0.499 + 0.866i)9-s + (−0.499 − 0.866i)12-s − i·13-s + (0.499 + 0.866i)16-s + (1.36 + 0.366i)19-s + (0.866 + 0.5i)25-s + 0.999i·27-s + (−0.366 − 1.36i)31-s − 0.999i·36-s + (1.36 + 0.366i)37-s + (0.5 − 0.866i)39-s + 0.999i·48-s + (−0.5 + 0.866i)52-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)3-s + (−0.866 − 0.5i)4-s + (0.499 + 0.866i)9-s + (−0.499 − 0.866i)12-s − i·13-s + (0.499 + 0.866i)16-s + (1.36 + 0.366i)19-s + (0.866 + 0.5i)25-s + 0.999i·27-s + (−0.366 − 1.36i)31-s − 0.999i·36-s + (1.36 + 0.366i)37-s + (0.5 − 0.866i)39-s + 0.999i·48-s + (−0.5 + 0.866i)52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.334004760\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.334004760\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + iT \) |
good | 2 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 5 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 37 | \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + iT^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 - iT^{2} \) |
| 73 | \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 97 | \( 1 + (-1 + i)T - iT^{2} \) |
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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
−9.477341338259458893306394282814, −8.760607957317663891197463693088, −7.956339498961771084502262398126, −7.40476030525170037811442556848, −5.97925185164848238615695661730, −5.27550245125876390201253947971, −4.48906863068299185180883496104, −3.59161207848206160350047328400, −2.74672940440215880259814618717, −1.24684666355824255855974736517,
1.22806627163669616304146127096, 2.64783293934053103488142933070, 3.44671643779255641204094163171, 4.33857934541090367550070505772, 5.15394308420142615881614614197, 6.43564896665599533813345617299, 7.22087102442766406190290067977, 7.86386597940860429570980764551, 8.667809404409515792891162518130, 9.254176189380290390276504021849