L(s) = 1 | + (0.965 + 0.258i)2-s + (0.707 + 0.707i)3-s + (0.500 + 0.866i)6-s + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s + 1.00i·9-s + (−0.258 + 0.965i)13-s + (0.866 − 0.500i)14-s + (−0.5 − 0.866i)16-s + (0.965 + 0.258i)17-s + (−0.258 + 0.965i)18-s + (0.866 + 0.5i)19-s + 1.00·21-s − 0.999i·24-s + (−0.499 + 0.866i)26-s + (−0.707 + 0.707i)27-s + ⋯ |
L(s) = 1 | + (0.965 + 0.258i)2-s + (0.707 + 0.707i)3-s + (0.500 + 0.866i)6-s + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s + 1.00i·9-s + (−0.258 + 0.965i)13-s + (0.866 − 0.500i)14-s + (−0.5 − 0.866i)16-s + (0.965 + 0.258i)17-s + (−0.258 + 0.965i)18-s + (0.866 + 0.5i)19-s + 1.00·21-s − 0.999i·24-s + (−0.499 + 0.866i)26-s + (−0.707 + 0.707i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.144694092\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.144694092\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 2 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 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 | \( 1 + iT^{2} \) |
| 29 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 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 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + 2T + T^{2} \) |
| 73 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
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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
−9.864101953592230753143292206540, −8.972536880883109640724116207594, −8.020520536251053597589218834887, −7.38804618673787384776763766208, −6.29589172712385127514043180874, −5.26036485602424613262808798955, −4.68094067830853605853580437579, −3.88627037559105298383878259479, −3.23924089804624685617367967777, −1.72536222333760870651217469146,
1.54377635240555344170191146008, 2.88977262446882646130614433621, 3.18708264890501264989660077222, 4.53028499009629204855373191096, 5.38155020608133063839802400433, 5.97872240584779656472603646679, 7.30517919104305582803904121734, 7.893323417589974983944957441709, 8.700647397858995282517722512812, 9.274670417507325236327233620610