L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.965 − 0.258i)3-s + (0.500 + 0.866i)6-s + (0.258 − 0.965i)7-s + (−0.707 + 0.707i)8-s + (0.866 + 0.499i)9-s + (0.965 + 0.258i)13-s + (−0.866 + 0.500i)14-s + 1.00·16-s + (0.965 − 0.258i)17-s + (−0.258 − 0.965i)18-s + (0.866 − 0.5i)19-s + (−0.499 + 0.866i)21-s + (0.866 − 0.5i)24-s + (−0.500 − 0.866i)26-s + (−0.707 − 0.707i)27-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.965 − 0.258i)3-s + (0.500 + 0.866i)6-s + (0.258 − 0.965i)7-s + (−0.707 + 0.707i)8-s + (0.866 + 0.499i)9-s + (0.965 + 0.258i)13-s + (−0.866 + 0.500i)14-s + 1.00·16-s + (0.965 − 0.258i)17-s + (−0.258 − 0.965i)18-s + (0.866 − 0.5i)19-s + (−0.499 + 0.866i)21-s + (0.866 − 0.5i)24-s + (−0.500 − 0.866i)26-s + (−0.707 − 0.707i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5982760658\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5982760658\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.965 + 0.258i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.258 + 0.965i)T \) |
good | 2 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 53 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 59 | \( 1 + iT - T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 71 | \( 1 + 2T + T^{2} \) |
| 73 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 - iT - T^{2} \) |
| 83 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
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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
−9.603319138827349392277281208194, −8.744087307891417081145011220106, −7.70022553231134499543271460783, −7.07438901958422040681689379389, −6.02071068137581853284546568107, −5.35284685739647334381662094454, −4.33607874120362633048353714018, −3.16003554654582166684090138636, −1.62785111165918555781214206912, −0.855545169668131933687633899687,
1.25884491901131651590652254621, 3.09969212426919849260878773271, 4.05167961364493317466960213993, 5.42284426143356360275084181331, 5.80431984061564972039347097281, 6.64909357074799859776883621271, 7.52511449678132289602629026028, 8.327132400615418582722418772737, 8.940947355638329889183173267826, 9.875493879717417614409662989809