L(s) = 1 | + 2.44i·5-s − 3.46·7-s + 8.48i·11-s − 10.3·13-s − 12.7i·17-s − 4·19-s − 34.2i·23-s + 19·25-s − 46.5i·29-s + 38.1·31-s − 8.48i·35-s + 27.7·37-s + 29.6i·41-s + 56·43-s + 63.6i·47-s + ⋯ |
L(s) = 1 | + 0.489i·5-s − 0.494·7-s + 0.771i·11-s − 0.799·13-s − 0.748i·17-s − 0.210·19-s − 1.49i·23-s + 0.760·25-s − 1.60i·29-s + 1.22·31-s − 0.242i·35-s + 0.748·37-s + 0.724i·41-s + 1.30·43-s + 1.35i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.570828072\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.570828072\) |
\(L(2)\) |
|
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 | \( 1 - 2.44iT - 25T^{2} \) |
| 7 | \( 1 + 3.46T + 49T^{2} \) |
| 11 | \( 1 - 8.48iT - 121T^{2} \) |
| 13 | \( 1 + 10.3T + 169T^{2} \) |
| 17 | \( 1 + 12.7iT - 289T^{2} \) |
| 19 | \( 1 + 4T + 361T^{2} \) |
| 23 | \( 1 + 34.2iT - 529T^{2} \) |
| 29 | \( 1 + 46.5iT - 841T^{2} \) |
| 31 | \( 1 - 38.1T + 961T^{2} \) |
| 37 | \( 1 - 27.7T + 1.36e3T^{2} \) |
| 41 | \( 1 - 29.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 56T + 1.84e3T^{2} \) |
| 47 | \( 1 - 63.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 71.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 101. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 69.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 28T + 4.48e3T^{2} \) |
| 71 | \( 1 + 53.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 20T + 5.32e3T^{2} \) |
| 79 | \( 1 - 72.7T + 6.24e3T^{2} \) |
| 83 | \( 1 - 76.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 55.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 8T + 9.40e3T^{2} \) |
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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
−9.137344653439902431042014604247, −8.001503555971082578942617367063, −7.42054450129020065310818798875, −6.56013418201947064974022330829, −6.06021007049268191110787673829, −4.70247615959883990236831686344, −4.33900252176644340000439463241, −2.79415156934707671431108711426, −2.51154941481793096695235016888, −0.810542913683926016881312822963,
0.51612905327263631671059588161, 1.70885930161060218508124616812, 2.97509715014874881185322111811, 3.72042111279777267969323733992, 4.82032392940589096702291320367, 5.50390331565036000013617277591, 6.37606817838134697049906085698, 7.14071169635474377019982652697, 8.043333761522213638009779883413, 8.721750898984348775796059222789