L(s) = 1 | + 1.07i·5-s − 7.21·7-s + 16.3i·11-s − 21.6·13-s + 18.9i·17-s + 17.0·19-s − 1.11i·23-s + 23.8·25-s + 29.4i·29-s + 5.63·31-s − 7.75i·35-s + 17.0·37-s + 27.4i·41-s − 52.3·43-s − 64.5i·47-s + ⋯ |
L(s) = 1 | + 0.214i·5-s − 1.03·7-s + 1.48i·11-s − 1.66·13-s + 1.11i·17-s + 0.897·19-s − 0.0485i·23-s + 0.953·25-s + 1.01i·29-s + 0.181·31-s − 0.221i·35-s + 0.460·37-s + 0.669i·41-s − 1.21·43-s − 1.37i·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
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.01039040347\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01039040347\) |
\(L(2)\) |
|
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 | \( 1 - 1.07iT - 25T^{2} \) |
| 7 | \( 1 + 7.21T + 49T^{2} \) |
| 11 | \( 1 - 16.3iT - 121T^{2} \) |
| 13 | \( 1 + 21.6T + 169T^{2} \) |
| 17 | \( 1 - 18.9iT - 289T^{2} \) |
| 19 | \( 1 - 17.0T + 361T^{2} \) |
| 23 | \( 1 + 1.11iT - 529T^{2} \) |
| 29 | \( 1 - 29.4iT - 841T^{2} \) |
| 31 | \( 1 - 5.63T + 961T^{2} \) |
| 37 | \( 1 - 17.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 27.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 52.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 64.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 35.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 56.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 69.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + 69.3T + 4.48e3T^{2} \) |
| 71 | \( 1 + 98.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 37.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 127.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 7.75iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 76.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 4.84T + 9.40e3T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.611678271498321493221847733536, −7.58273822840277493443074738680, −6.99456375954023348573946667628, −6.45945785539901496010804933732, −5.24353349350759372640448863035, −4.65919230578376984906114334634, −3.52364515146807392209260988985, −2.69694242440984861445954629405, −1.67569645655675776812091963598, −0.00297910643702144398901182126,
0.919291014686476275688669939781, 2.72941827871071753548409476662, 3.04216618880880090458603036595, 4.30106688334742353294578415008, 5.23446577566429324223939878456, 5.90333747319912877916185392263, 6.85727104844606494072679250706, 7.45459835354108630146602504498, 8.372380554922185592081025734474, 9.242193844733833311249652614124