L(s) = 1 | + (0.866 − 1.5i)2-s + (−0.5 − 0.866i)4-s + (1.73 − 3i)5-s + 1.73·8-s + (−3 − 5.19i)10-s + (1.73 + 3i)11-s − 2·13-s + (2.49 − 4.33i)16-s + (−1.73 − 3i)17-s + (−2 + 3.46i)19-s − 3.46·20-s + 6·22-s + (−1.73 + 3i)23-s + (−3.5 − 6.06i)25-s + (−1.73 + 3i)26-s + ⋯ |
L(s) = 1 | + (0.612 − 1.06i)2-s + (−0.250 − 0.433i)4-s + (0.774 − 1.34i)5-s + 0.612·8-s + (−0.948 − 1.64i)10-s + (0.522 + 0.904i)11-s − 0.554·13-s + (0.624 − 1.08i)16-s + (−0.420 − 0.727i)17-s + (−0.458 + 0.794i)19-s − 0.774·20-s + 1.27·22-s + (−0.361 + 0.625i)23-s + (−0.700 − 1.21i)25-s + (−0.339 + 0.588i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.36065 - 1.78855i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36065 - 1.78855i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.866 + 1.5i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.73 + 3i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.73 - 3i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (1.73 + 3i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.73 - 3i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (3.46 - 6i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.46 + 6i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.46 - 6i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + (-7 - 12.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-1.73 + 3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−11.07597797889966436151275821958, −9.762108333841028322958995533774, −9.577924777195509527047162479013, −8.276255283095500725974311184833, −7.15493606615762902651882748294, −5.70767997243788889678556460443, −4.76788602342978370105735376727, −4.05027739523580402681731346377, −2.39308240494693348068871152938, −1.42585812511258605735415408409,
2.17410195544381373307158306863, 3.57164751626833891532980491553, 4.89009565842659254257570699128, 6.11017428410712228794791537419, 6.45490217576550286973194696935, 7.30784016852034462052284285322, 8.435306804017051460939841793516, 9.608106847756638524911174073172, 10.71931607229873905240695053482, 11.02090591274987117362807617047