L(s) = 1 | + (−0.258 + 0.965i)3-s + (0.5 − 0.866i)4-s + (0.258 − 0.448i)7-s + (−0.866 − 0.499i)9-s + (0.707 + 0.707i)12-s + (0.707 + 0.707i)13-s + (−0.499 − 0.866i)16-s + (1.86 − 0.5i)19-s + (0.366 + 0.366i)21-s + (0.707 − 0.707i)27-s + (−0.258 − 0.448i)28-s + (−1 + i)31-s + (−0.866 + 0.5i)36-s + (−0.707 − 1.22i)37-s + (−0.866 + 0.500i)39-s + ⋯ |
L(s) = 1 | + (−0.258 + 0.965i)3-s + (0.5 − 0.866i)4-s + (0.258 − 0.448i)7-s + (−0.866 − 0.499i)9-s + (0.707 + 0.707i)12-s + (0.707 + 0.707i)13-s + (−0.499 − 0.866i)16-s + (1.86 − 0.5i)19-s + (0.366 + 0.366i)21-s + (0.707 − 0.707i)27-s + (−0.258 − 0.448i)28-s + (−1 + i)31-s + (−0.866 + 0.5i)36-s + (−0.707 − 1.22i)37-s + (−0.866 + 0.500i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.082485921\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.082485921\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.258 - 0.965i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-0.707 - 0.707i)T \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (-1.86 + 0.5i)T + (0.866 - 0.5i)T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (1 - i)T - iT^{2} \) |
| 37 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 43 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (1.67 - 0.965i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 73 | \( 1 + 1.93iT - T^{2} \) |
| 79 | \( 1 - 1.73iT - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 97 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
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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
−10.32670938599741472908552523663, −9.415815086439806315832038786835, −8.964984020973385944997042425707, −7.56070878675434033248299133634, −6.73174145664706885826633712543, −5.72444688408558293112881316651, −5.09489943760074772048771734869, −4.07406308127432204756339950451, −2.97113717335325304440658895552, −1.33911536005224396299401071653,
1.54736595501094255500192389735, 2.76261547294728138366651111106, 3.64310317762168944937690530596, 5.27813290945171049426689562941, 5.95892522046567585140658812211, 6.98963651393750828481915190980, 7.68564593431478014158403037415, 8.270097724890084470484961782205, 9.121296920269636136024562877150, 10.41225214136087386503751837643