| L(s) = 1 | − i·3-s + 1.41i·5-s − 9-s − 4i·11-s + 1.41i·13-s + 1.41·15-s + 4i·19-s + 5.65·23-s + 2.99·25-s + i·27-s + 7.07i·29-s − 5.65·31-s − 4·33-s + 4.24i·37-s + 1.41·39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.632i·5-s − 0.333·9-s − 1.20i·11-s + 0.392i·13-s + 0.365·15-s + 0.917i·19-s + 1.17·23-s + 0.599·25-s + 0.192i·27-s + 1.31i·29-s − 1.01·31-s − 0.696·33-s + 0.697i·37-s + 0.226·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.776599895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.776599895\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| good | 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 - 1.41iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 - 7.07iT - 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 - 4.24iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 - 1.41iT - 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 + 12.7iT - 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 16.9T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−8.705252639560752809143479448092, −7.932521820642763385710505693519, −7.05185680458004524462707650024, −6.63183437252632265182739170597, −5.72312426342127197195862597049, −5.05673458612628283902143950543, −3.66935993015824193499090218273, −3.15154381091793796501139302291, −2.05376380654146579220679512052, −0.892128417132368433853562068890,
0.74891855722496188045815864050, 2.12923186555589464888666718699, 3.09546902043907922974688174885, 4.21603681554694101490611709564, 4.79611388457732846109673928860, 5.41286292972813491643528626143, 6.44176612916885200996774478781, 7.29419756419364658092534566452, 7.961103448082896735541627097759, 8.973413459061125589091857271312
