L(s) = 1 | + (1.40 + 1.01i)3-s + (1.61 − 1.17i)4-s + (−1.35 − 1.86i)7-s + (0.927 + 2.85i)9-s + 3.46·12-s + (6.00 − 1.95i)13-s + (1.23 − 3.80i)16-s + (−5.06 + 6.97i)19-s − 4.00i·21-s + (−4.04 − 2.93i)25-s + (−1.60 + 4.94i)27-s + (−4.39 − 1.42i)28-s + (0.535 + 1.64i)31-s + (4.85 + 3.52i)36-s + (−4.20 + 3.05i)37-s + ⋯ |
L(s) = 1 | + (0.809 + 0.587i)3-s + (0.809 − 0.587i)4-s + (−0.513 − 0.706i)7-s + (0.309 + 0.951i)9-s + 1.00·12-s + (1.66 − 0.541i)13-s + (0.309 − 0.951i)16-s + (−1.16 + 1.60i)19-s − 0.873i·21-s + (−0.809 − 0.587i)25-s + (−0.309 + 0.951i)27-s + (−0.830 − 0.269i)28-s + (0.0961 + 0.295i)31-s + (0.809 + 0.587i)36-s + (−0.691 + 0.502i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0138i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.99662 - 0.0138741i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.99662 - 0.0138741i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.40 - 1.01i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-1.61 + 1.17i)T^{2} \) |
| 5 | \( 1 + (4.04 + 2.93i)T^{2} \) |
| 7 | \( 1 + (1.35 + 1.86i)T + (-2.16 + 6.65i)T^{2} \) |
| 13 | \( 1 + (-6.00 + 1.95i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (5.06 - 6.97i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.535 - 1.64i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (4.20 - 3.05i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 1.69iT - 43T^{2} \) |
| 47 | \( 1 + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (9.81 + 3.18i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-7.78 - 10.7i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-0.588 + 0.191i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + (1.54 + 4.75i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.02508963407207415755861658178, −10.42719116453746110957907785812, −9.881516849266974991256999997949, −8.582603737128785752921989015159, −7.81739723227417578458938316570, −6.56842542247371709473487451727, −5.73237159526083871991142233325, −4.12545726433783186806020501170, −3.23156906920866213289670812530, −1.67976227728689706138237305400,
1.88820590257029206013540618069, 2.96922002006688536407797611589, 3.99587305591651619310487670796, 6.07578929823513212925195431536, 6.66072685647908960114225186338, 7.67564466430078355353044315916, 8.743197624955403686884325904110, 9.128860350824724932543048682663, 10.71582619830076889694866460809, 11.54359135228133541169870737492