L(s) = 1 | − 8i·5-s − 10i·13-s − 16·17-s − 39·25-s − 40i·29-s + 70i·37-s − 80·41-s + 49·49-s − 56i·53-s − 22i·61-s − 80·65-s − 110·73-s + 128i·85-s + 160·89-s − 130·97-s + ⋯ |
L(s) = 1 | − 1.60i·5-s − 0.769i·13-s − 0.941·17-s − 1.56·25-s − 1.37i·29-s + 1.89i·37-s − 1.95·41-s + 0.999·49-s − 1.05i·53-s − 0.360i·61-s − 1.23·65-s − 1.50·73-s + 1.50i·85-s + 1.79·89-s − 1.34·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4287745754\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4287745754\) |
\(L(2)\) |
|
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 | \( 1 + 8iT - 25T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 11 | \( 1 + 121T^{2} \) |
| 13 | \( 1 + 10iT - 169T^{2} \) |
| 17 | \( 1 + 16T + 289T^{2} \) |
| 19 | \( 1 + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 + 40iT - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 70iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 80T + 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 + 56iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 3.48e3T^{2} \) |
| 61 | \( 1 + 22iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 110T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 - 160T + 7.92e3T^{2} \) |
| 97 | \( 1 + 130T + 9.40e3T^{2} \) |
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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
−8.410717922423940163256048948958, −7.85270983954925281257458878700, −6.73710319460723332686557801606, −5.87291774803100614173443823406, −5.01722402598296722380591871320, −4.52149573289662922536254066216, −3.48727403363898510937007883629, −2.20624476668426273428652406136, −1.12192597085330551879615934889, −0.10680510751916947564980557105,
1.77237310731648065597035701029, 2.65013416968999020829653281068, 3.52158219445314296362992092321, 4.36135707365332285886039434454, 5.50588378388715542281210878335, 6.40632977768699523168355530942, 7.00292305791717329107484390345, 7.46170853131590429054819953335, 8.663925471996682732216225294959, 9.232209136851508745226665343860