L(s) = 1 | − 9i·3-s + 66i·5-s − 176·7-s − 81·9-s + 60i·11-s − 658i·13-s + 594·15-s − 414·17-s + 956i·19-s + 1.58e3i·21-s − 600·23-s − 1.23e3·25-s + 729i·27-s + 5.57e3i·29-s − 3.59e3·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 1.18i·5-s − 1.35·7-s − 0.333·9-s + 0.149i·11-s − 1.07i·13-s + 0.681·15-s − 0.347·17-s + 0.607i·19-s + 0.783i·21-s − 0.236·23-s − 0.393·25-s + 0.192i·27-s + 1.23i·29-s − 0.671·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.9210070900\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9210070900\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 9iT \) |
good | 5 | \( 1 - 66iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 176T + 1.68e4T^{2} \) |
| 11 | \( 1 - 60iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 658iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 414T + 1.41e6T^{2} \) |
| 19 | \( 1 - 956iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 600T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.57e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 3.59e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.45e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.91e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.33e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.96e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.12e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.63e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 3.10e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 1.68e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 6.12e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.55e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.44e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.46e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 3.27e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.66e5T + 8.58e9T^{2} \) |
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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
−9.627956951777188351383610021364, −8.515517746779231346042023018420, −7.54579789605196459641968534559, −6.72944144106114820850657613817, −6.28904430562815054720158504414, −5.19383797929831271627884937579, −3.40733619228593721953675994441, −3.13640398197468394010441647131, −1.86630436128317546232919389117, −0.30853220453386472607298916348,
0.58378379597705921947381682142, 2.05755516552421735573510736203, 3.37239726725616879032834180772, 4.26172411170301245831151030338, 5.08567583040468785964965990746, 6.14281496370255785309225603206, 6.91194827726228191032372458387, 8.225918206651193673344884081673, 9.090656248938958834233976442172, 9.461271113192129148826399846248