L(s) = 1 | + 3.78i·5-s − 0.953·7-s − 11.6i·11-s − 8.69·13-s − 4.73i·17-s + 29.2·19-s − 2.69i·23-s + 10.6·25-s − 21.7i·29-s + 22.5·31-s − 3.60i·35-s + 24.0·37-s + 19.7i·41-s + 49.0·43-s + 59.3i·47-s + ⋯ |
L(s) = 1 | + 0.756i·5-s − 0.136·7-s − 1.06i·11-s − 0.668·13-s − 0.278i·17-s + 1.54·19-s − 0.117i·23-s + 0.427·25-s − 0.748i·29-s + 0.728·31-s − 0.103i·35-s + 0.651·37-s + 0.482i·41-s + 1.13·43-s + 1.26i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.812919950\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.812919950\) |
\(L(2)\) |
|
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 | \( 1 - 3.78iT - 25T^{2} \) |
| 7 | \( 1 + 0.953T + 49T^{2} \) |
| 11 | \( 1 + 11.6iT - 121T^{2} \) |
| 13 | \( 1 + 8.69T + 169T^{2} \) |
| 17 | \( 1 + 4.73iT - 289T^{2} \) |
| 19 | \( 1 - 29.2T + 361T^{2} \) |
| 23 | \( 1 + 2.69iT - 529T^{2} \) |
| 29 | \( 1 + 21.7iT - 841T^{2} \) |
| 31 | \( 1 - 22.5T + 961T^{2} \) |
| 37 | \( 1 - 24.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 19.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 49.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 59.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 48.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 12.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 111.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 56.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 35.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 144.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 101. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 19.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 27T + 9.40e3T^{2} \) |
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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
−9.884811185040334646086579541187, −9.316053514602441288878131788663, −8.136992878072810747208170559004, −7.43922488117597927103861447612, −6.50084010986036581936075913761, −5.69546052260534086523642005615, −4.61153733205275525048009942313, −3.29640202470257446429907913390, −2.63212882973310192314935502732, −0.827507057061225326314086621153,
0.939299003074187441914200438649, 2.27156273922677569525150618073, 3.59774878977814720729834345241, 4.80367892289515334690166928694, 5.29815931199080916438167394875, 6.62684707763213237899820448511, 7.43695769368230045059136857494, 8.252014905731431873679010012157, 9.338472913140382267219944461382, 9.732885595764637189194969842799