L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.866 + 1.5i)5-s − 0.999i·8-s + (1.5 − 0.866i)10-s + (−0.866 + 0.5i)11-s + (−0.5 + 0.866i)16-s − 1.73·20-s + 0.999·22-s + (−1 − 1.73i)25-s + i·29-s + (−1.5 + 0.866i)31-s + (0.866 − 0.499i)32-s + (1.49 + 0.866i)40-s + (−0.866 − 0.499i)44-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.866 + 1.5i)5-s − 0.999i·8-s + (1.5 − 0.866i)10-s + (−0.866 + 0.5i)11-s + (−0.5 + 0.866i)16-s − 1.73·20-s + 0.999·22-s + (−1 − 1.73i)25-s + i·29-s + (−1.5 + 0.866i)31-s + (0.866 − 0.499i)32-s + (1.49 + 0.866i)40-s + (−0.866 − 0.499i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1752182745\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1752182745\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - iT - T^{2} \) |
| 31 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + 1.73T + T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + 1.73iT - T^{2} \) |
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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
−9.132583556003926096650784734231, −8.308759309458413769922156777222, −7.63689326303382885471010980385, −7.10077849656936229823598509566, −6.62794444712785041153857536852, −5.40325792910807685579913033669, −4.19618467312770418752699434175, −3.36216327225643675048503068124, −2.80345752791468587936824401929, −1.81512168043010021678651177499,
0.14187585088491212629746735570, 1.26662938912428378822387550457, 2.49854503131194138573829266227, 3.85595229068464594085815814716, 4.71040650775833899404681191272, 5.48225692302743293197348066355, 6.01678690582957789126432851271, 7.32164675116366659269509432260, 7.72228523602223165996687919850, 8.396882211619231081979559630112