L(s) = 1 | + 2.60i·7-s − 3·9-s + 6.60i·11-s − 3.60·13-s + 7.21·17-s − 1.39i·19-s + 5·25-s − 7.21·29-s + 10.6i·31-s − 5.39i·47-s + 0.211·49-s − 2·53-s − 11.8i·59-s + 6·61-s − 7.81i·63-s + ⋯ |
L(s) = 1 | + 0.984i·7-s − 9-s + 1.99i·11-s − 1.00·13-s + 1.74·17-s − 0.319i·19-s + 25-s − 1.33·29-s + 1.90i·31-s − 0.786i·47-s + 0.0301·49-s − 0.274·53-s − 1.53i·59-s + 0.768·61-s − 0.984i·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.774458 + 0.774458i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.774458 + 0.774458i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 + 3.60T \) |
good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 2.60iT - 7T^{2} \) |
| 11 | \( 1 - 6.60iT - 11T^{2} \) |
| 17 | \( 1 - 7.21T + 17T^{2} \) |
| 19 | \( 1 + 1.39iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 7.21T + 29T^{2} \) |
| 31 | \( 1 - 10.6iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 5.39iT - 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 11.8iT - 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 14.6iT - 67T^{2} \) |
| 71 | \( 1 + 15.8iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 3.81iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−11.69329853030028577295600433285, −10.39183781127004033472534018583, −9.608050738002508498892501317916, −8.820540867512384582929234010368, −7.69513306712950085322864099729, −6.87136780288255110084585463440, −5.46692934704406891174584627231, −4.91638418915529107947834269656, −3.18012459969823296968349578512, −2.05188814638214753610191041462,
0.70528542293496420345448771132, 2.90091462020714430126690557653, 3.81429332898554409211675014827, 5.38996464889462365729108582911, 6.04800720993374050932465652675, 7.46834332425262929018324630645, 8.106613333000413007642276749409, 9.170146470494094362513778209799, 10.16782107604686935506194657565, 11.06267973651422405629713947689