L(s) = 1 | + 1.73i·3-s − 6.84·5-s + 3.94·7-s − 2.99·9-s + 15.1·11-s + 1.55i·13-s − 11.8i·15-s + 2.94·17-s + (−16.0 + 10.1i)19-s + 6.84i·21-s + 20.2·23-s + 21.9·25-s − 5.19i·27-s − 4.45i·29-s − 3.63i·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 1.36·5-s + 0.564·7-s − 0.333·9-s + 1.37·11-s + 0.119i·13-s − 0.790i·15-s + 0.173·17-s + (−0.846 + 0.533i)19-s + 0.325i·21-s + 0.879·23-s + 0.876·25-s − 0.192i·27-s − 0.153i·29-s − 0.117i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.533 - 0.846i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.533 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.130723095\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.130723095\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 19 | \( 1 + (16.0 - 10.1i)T \) |
good | 5 | \( 1 + 6.84T + 25T^{2} \) |
| 7 | \( 1 - 3.94T + 49T^{2} \) |
| 11 | \( 1 - 15.1T + 121T^{2} \) |
| 13 | \( 1 - 1.55iT - 169T^{2} \) |
| 17 | \( 1 - 2.94T + 289T^{2} \) |
| 23 | \( 1 - 20.2T + 529T^{2} \) |
| 29 | \( 1 + 4.45iT - 841T^{2} \) |
| 31 | \( 1 + 3.63iT - 961T^{2} \) |
| 37 | \( 1 + 17.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 76.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 26.7T + 1.84e3T^{2} \) |
| 47 | \( 1 - 48.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 49.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 97.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 105.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 129. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 130. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 15.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 113. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 62.5T + 6.88e3T^{2} \) |
| 89 | \( 1 + 134. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 116. iT - 9.40e3T^{2} \) |
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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
−10.27213808808909136640861197845, −9.186220791259614237668022633840, −8.555237107738949960620517515479, −7.75319738217909089494654841743, −6.87451096542357479657054067820, −5.81595794716274159995794362313, −4.43605813688710649457076250764, −4.15340272051230303645843218482, −3.04133990746826869843488165152, −1.27387948216681482950063526737,
0.41463494131446366738855331423, 1.71349395699913468661594750900, 3.26332477327570001892544171571, 4.14613109877954760369656939181, 5.06965241654045563934009781663, 6.45375505085714545521990953147, 7.07749738960203239368320659242, 7.939634945950074142143827133235, 8.604243109804018680884269608041, 9.367386135605333248975412064241