L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.707 − 1.22i)5-s + 0.999·8-s + (0.707 + 1.22i)10-s + 1.41·13-s + (−0.5 + 0.866i)16-s + (−0.707 − 1.22i)17-s − 1.41·20-s + (−0.499 − 0.866i)25-s + (−0.707 + 1.22i)26-s − 2·29-s + (−0.499 − 0.866i)32-s + 1.41·34-s + (0.707 − 1.22i)40-s + 1.41·41-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.707 − 1.22i)5-s + 0.999·8-s + (0.707 + 1.22i)10-s + 1.41·13-s + (−0.5 + 0.866i)16-s + (−0.707 − 1.22i)17-s − 1.41·20-s + (−0.499 − 0.866i)25-s + (−0.707 + 1.22i)26-s − 2·29-s + (−0.499 − 0.866i)32-s + 1.41·34-s + (0.707 − 1.22i)40-s + 1.41·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9613922268\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9613922268\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - 1.41T + T^{2} \) |
| 17 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + 2T + T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - 1.41T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + 1.41T + T^{2} \) |
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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
−9.229641951583521366417247924755, −8.798183215751483641580790524763, −7.987296331461923099148164036296, −7.10366130523002705997653649497, −6.15735368025110331681855042333, −5.53641733325952376367023765449, −4.82107656251439584939117241219, −3.86118595625707887904445289356, −2.04660180186693198617227803882, −0.947307761203043917555041474569,
1.59659538639705878876861769756, 2.44210338167708534506680586034, 3.50391329763465637679242450302, 4.12641046610760009615627387946, 5.65910093255353624535780162829, 6.35973050890619636035788423990, 7.23483675241046015705311245436, 8.124088053501650816192385652021, 8.939690708279310740518563832350, 9.596853226029443924982534806085