L(s) = 1 | + (0.965 − 0.258i)2-s + (−0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + (−0.866 − 0.5i)9-s + (−0.5 − 0.866i)11-s + (−1.22 − 1.22i)13-s + (−0.866 − 0.500i)14-s + (−0.5 + 0.866i)16-s + (−0.965 − 0.258i)18-s + (−0.707 − 0.707i)22-s + (−1.49 − 0.866i)26-s + (1.5 − 0.866i)31-s + (−0.707 + 0.707i)43-s + 1.00i·49-s + 0.999·56-s + ⋯ |
L(s) = 1 | + (0.965 − 0.258i)2-s + (−0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + (−0.866 − 0.5i)9-s + (−0.5 − 0.866i)11-s + (−1.22 − 1.22i)13-s + (−0.866 − 0.500i)14-s + (−0.5 + 0.866i)16-s + (−0.965 − 0.258i)18-s + (−0.707 − 0.707i)22-s + (−1.49 − 0.866i)26-s + (1.5 − 0.866i)31-s + (−0.707 + 0.707i)43-s + 1.00i·49-s + 0.999·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7854391537\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7854391537\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 3 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 13 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 17 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 47 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + (-0.448 + 1.67i)T + (-0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 89 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−9.085816828879446125524908985165, −8.242754655539004990532037501001, −7.59163724824206049907519499586, −6.37364576980478339575752786690, −5.77339305465967460838322055786, −4.99881081456323419871825617442, −4.06203567003385870650670516207, −2.98805141044967910502260477058, −2.83716118017684940680562085020, −0.38508988072173810572992009311,
2.24872366267873055708789953479, 2.94434793050898216193179563446, 4.13917700618545449922487999298, 5.02392984433419839636194908108, 5.40565732882500622977575993880, 6.53758228801142803685325464144, 6.94041878306722539941046356256, 8.144336268775481078745644761259, 8.985388047601222815825176325207, 9.699387006315549793398576842772