L(s) = 1 | − 2i·2-s − 4·4-s + 13i·7-s + 8i·8-s − 30·11-s − 61i·13-s + 26·14-s + 16·16-s − 12i·17-s + 49·19-s + 60i·22-s + 18i·23-s − 122·26-s − 52i·28-s + 186·29-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.701i·7-s + 0.353i·8-s − 0.822·11-s − 1.30i·13-s + 0.496·14-s + 0.250·16-s − 0.171i·17-s + 0.591·19-s + 0.581i·22-s + 0.163i·23-s − 0.920·26-s − 0.350i·28-s + 1.19·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.1849161736\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1849161736\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 13iT - 343T^{2} \) |
| 11 | \( 1 + 30T + 1.33e3T^{2} \) |
| 13 | \( 1 + 61iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 12iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 49T + 6.85e3T^{2} \) |
| 23 | \( 1 - 18iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 186T + 2.43e4T^{2} \) |
| 31 | \( 1 + 160T + 2.97e4T^{2} \) |
| 37 | \( 1 - 91iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 378T + 6.89e4T^{2} \) |
| 43 | \( 1 + 268iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 144iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 570iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 204T + 2.05e5T^{2} \) |
| 61 | \( 1 + 877T + 2.26e5T^{2} \) |
| 67 | \( 1 - 187iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 606T + 3.57e5T^{2} \) |
| 73 | \( 1 - 431iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.15e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 102iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 984T + 7.04e5T^{2} \) |
| 97 | \( 1 - 265iT - 9.12e5T^{2} \) |
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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
−9.564761769758543862653475748844, −8.790808247148220346110217456488, −8.007365240798604708540257608186, −7.26340383987558196194507986400, −5.81714220743060773863603325223, −5.39446144541566994538082280386, −4.34518000464995601005167117026, −3.06568724449418138313411360358, −2.57867646435804228174035187787, −1.17967776171013015864471717909,
0.04564443256450501203090249758, 1.40439505451726550675672085872, 2.82723593440276074018956128534, 4.06194061174577514478596781255, 4.71352126708001978741484292084, 5.73182144320499437276733258103, 6.60371642993232618353889771195, 7.35681374912598594798359376045, 7.970677178766323666608083303722, 8.953832419026667716413265868582