L(s) = 1 | + 1.41i·5-s − 24·13-s − 32.5i·17-s + 23·25-s − 1.41i·29-s − 70·37-s − 69.2i·41-s − 49·49-s − 103. i·53-s + 22·61-s − 33.9i·65-s − 96·73-s + 46·85-s + 168. i·89-s − 144·97-s + ⋯ |
L(s) = 1 | + 0.282i·5-s − 1.84·13-s − 1.91i·17-s + 0.920·25-s − 0.0487i·29-s − 1.89·37-s − 1.69i·41-s − 0.999·49-s − 1.94i·53-s + 0.360·61-s − 0.522i·65-s − 1.31·73-s + 0.541·85-s + 1.89i·89-s − 1.48·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7270940938\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7270940938\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 1.41iT - 25T^{2} \) |
| 7 | \( 1 + 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + 24T + 169T^{2} \) |
| 17 | \( 1 + 32.5iT - 289T^{2} \) |
| 19 | \( 1 + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 + 1.41iT - 841T^{2} \) |
| 31 | \( 1 + 961T^{2} \) |
| 37 | \( 1 + 70T + 1.36e3T^{2} \) |
| 41 | \( 1 + 69.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 + 103. iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 22T + 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 96T + 5.32e3T^{2} \) |
| 79 | \( 1 + 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 - 168. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 144T + 9.40e3T^{2} \) |
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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
−10.08494080208346222567750194276, −9.490732793872350910346938633563, −8.492482779520382743389369102100, −7.17005386975135278363224381530, −7.01136068515800159656906813508, −5.35731878198592717575944990797, −4.75715733647111035288795420517, −3.23057392603727289073195196204, −2.23723818144808513992793097267, −0.26322178507090570945095268943,
1.62764583852902386180500945494, 2.97858099974870590522382100617, 4.32790478626635867357995846067, 5.18651881231061320380005127871, 6.30591857902515335696342987297, 7.28870971762089548160984716158, 8.194102370999916383453735283636, 9.050155068314691025662537352192, 10.04729492218309958602155786294, 10.63811605184534116046486820346