L(s) = 1 | − 2.46i·5-s − 5.48i·7-s + 10.5·11-s + 8.96i·13-s + 16.2·17-s − 2.96·19-s − 1.46i·23-s + 18.9·25-s − 25.0i·29-s − 10.5i·31-s − 13.5·35-s + 16.9i·37-s − 29.0·41-s + 34.9·43-s + 86.1i·47-s + ⋯ |
L(s) = 1 | − 0.492i·5-s − 0.783i·7-s + 0.962·11-s + 0.689i·13-s + 0.955·17-s − 0.156·19-s − 0.0635i·23-s + 0.757·25-s − 0.865i·29-s − 0.339i·31-s − 0.385·35-s + 0.458i·37-s − 0.707·41-s + 0.813·43-s + 1.83i·47-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
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.328910038\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.328910038\) |
\(L(2)\) |
|
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 | \( 1 + 2.46iT - 25T^{2} \) |
| 7 | \( 1 + 5.48iT - 49T^{2} \) |
| 11 | \( 1 - 10.5T + 121T^{2} \) |
| 13 | \( 1 - 8.96iT - 169T^{2} \) |
| 17 | \( 1 - 16.2T + 289T^{2} \) |
| 19 | \( 1 + 2.96T + 361T^{2} \) |
| 23 | \( 1 + 1.46iT - 529T^{2} \) |
| 29 | \( 1 + 25.0iT - 841T^{2} \) |
| 31 | \( 1 + 10.5iT - 961T^{2} \) |
| 37 | \( 1 - 16.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 29.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 34.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 86.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 80.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 66.4T + 3.48e3T^{2} \) |
| 61 | \( 1 + 0.966iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 113.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 90.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 51.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 80.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 79.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + 142.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 45.8T + 9.40e3T^{2} \) |
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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
−8.799471404907864555631344215494, −7.943707954722174307503960431379, −7.19300396170066454011113204809, −6.44605938265641931218147525085, −5.62498257619972715367681965733, −4.48874426992059388680033182158, −4.06875966937822359349323543969, −2.95790931610458378514375765480, −1.57869897821058179077864820931, −0.75160039240003863071761584682,
0.933309086868771325246903035922, 2.16874663811597812984495556167, 3.16605739091386625737379380351, 3.87244565142522702250227500758, 5.19131909212863623016011693055, 5.67485034294769062448401780360, 6.72645331904863510953573454289, 7.19278037187575534779936105285, 8.389913941479089827927262106847, 8.737223313465888969287969644331