| L(s) = 1 | + (1.72 − 1.00i)2-s + (−0.482 + 2.96i)3-s + (1.97 − 3.47i)4-s + (−0.0274 − 0.0100i)5-s + (2.14 + 5.60i)6-s + (11.2 + 1.98i)7-s + (−0.0814 − 7.99i)8-s + (−8.53 − 2.85i)9-s + (−0.0575 + 0.0103i)10-s + (4.17 + 11.4i)11-s + (9.34 + 7.53i)12-s + (0.356 − 0.299i)13-s + (21.4 − 7.87i)14-s + (0.0428 − 0.0765i)15-s + (−8.18 − 13.7i)16-s + (−6.07 − 10.5i)17-s + ⋯ |
| L(s) = 1 | + (0.864 − 0.502i)2-s + (−0.160 + 0.986i)3-s + (0.494 − 0.869i)4-s + (−0.00549 − 0.00200i)5-s + (0.357 + 0.933i)6-s + (1.60 + 0.282i)7-s + (−0.0101 − 0.999i)8-s + (−0.948 − 0.317i)9-s + (−0.00575 + 0.00103i)10-s + (0.379 + 1.04i)11-s + (0.778 + 0.627i)12-s + (0.0274 − 0.0230i)13-s + (1.52 − 0.562i)14-s + (0.00285 − 0.00510i)15-s + (−0.511 − 0.859i)16-s + (−0.357 − 0.619i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0126i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.19376 - 0.0138358i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.19376 - 0.0138358i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.72 + 1.00i)T \) |
| 3 | \( 1 + (0.482 - 2.96i)T \) |
| good | 5 | \( 1 + (0.0274 + 0.0100i)T + (19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (-11.2 - 1.98i)T + (46.0 + 16.7i)T^{2} \) |
| 11 | \( 1 + (-4.17 - 11.4i)T + (-92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (-0.356 + 0.299i)T + (29.3 - 166. i)T^{2} \) |
| 17 | \( 1 + (6.07 + 10.5i)T + (-144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (14.4 + 8.34i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (43.8 - 7.72i)T + (497. - 180. i)T^{2} \) |
| 29 | \( 1 + (-0.0229 - 0.0192i)T + (146. + 828. i)T^{2} \) |
| 31 | \( 1 + (-41.6 + 7.35i)T + (903. - 328. i)T^{2} \) |
| 37 | \( 1 + (-11.7 - 20.3i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (3.36 - 2.82i)T + (291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-12.3 - 34.0i)T + (-1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (45.6 + 8.05i)T + (2.07e3 + 755. i)T^{2} \) |
| 53 | \( 1 - 10.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + (16.2 - 44.7i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-2.22 + 12.6i)T + (-3.49e3 - 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-44.1 - 52.5i)T + (-779. + 4.42e3i)T^{2} \) |
| 71 | \( 1 + (15.6 - 9.04i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-66.0 + 114. i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-40.9 + 48.8i)T + (-1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-51.0 + 60.8i)T + (-1.19e3 - 6.78e3i)T^{2} \) |
| 89 | \( 1 + (-36.4 + 63.0i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (113. - 41.3i)T + (7.20e3 - 6.04e3i)T^{2} \) |
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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
−13.68456149365130660191244523584, −11.96580644180402944821089552700, −11.62175788127807067949875811191, −10.47467236248344149267413972753, −9.542277474865173044700916829132, −8.060363657611121141258233836085, −6.21680151688888525631026967166, −4.84336407201790509775994662737, −4.25110658569511915956358833039, −2.18861367643681359307786409683,
1.95037247016095728436842522958, 4.07002012195754344249934714172, 5.57160814028808476932117165844, 6.53611584807644250921443204652, 8.036274884209441177253545087605, 8.296478505178230411266467088122, 10.89402138137490825730361211968, 11.60598666699987168743597697463, 12.48813273776220281283926427967, 13.83038554348116390423815171460
