L(s) = 1 | + (0.707 + 0.707i)3-s + 1.73i·11-s + (1.22 + 1.22i)17-s − 1.73·19-s + (0.707 − 0.707i)27-s + (−1.22 + 1.22i)33-s − 41-s + (1.41 + 1.41i)43-s − i·49-s + 1.73i·51-s + (−1.22 − 1.22i)57-s + (−0.707 + 0.707i)67-s + (1.22 − 1.22i)73-s + 1.00·81-s + (0.707 + 0.707i)83-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)3-s + 1.73i·11-s + (1.22 + 1.22i)17-s − 1.73·19-s + (0.707 − 0.707i)27-s + (−1.22 + 1.22i)33-s − 41-s + (1.41 + 1.41i)43-s − i·49-s + 1.73i·51-s + (−1.22 − 1.22i)57-s + (−0.707 + 0.707i)67-s + (1.22 − 1.22i)73-s + 1.00·81-s + (0.707 + 0.707i)83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.537507318\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.537507318\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 - 1.73iT - T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 19 | \( 1 + 1.73T + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 89 | \( 1 - iT - T^{2} \) |
| 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
−9.065068559313306818621856336092, −8.317377467178684788760215940235, −7.72692586606656199104936462002, −6.75710675404685100932215814951, −6.08415644969605254369043268184, −4.95395884749949029801514043122, −4.22206333604225556196704049747, −3.67494598080229290651240763701, −2.56337491938000895277093103153, −1.65481608587261929640668256165,
0.880895053263373324823330517697, 2.16174029248050337525541524261, 2.96938910282853536737491927868, 3.71032120465903672724342247658, 4.90347729965299097238663303607, 5.75142274661073388479179566591, 6.46824260738211671990522566115, 7.37868696026063083408716807689, 7.928682473655379003471348593741, 8.680965434493629555124480800243