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
−11.54359135228133541169870737492, −10.71582619830076889694866460809, −9.128860350824724932543048682663, −8.743197624955403686884325904110, −7.67564466430078355353044315916, −6.66072685647908960114225186338, −6.07578929823513212925195431536, −3.99587305591651619310487670796, −2.96922002006688536407797611589, −1.88820590257029206013540618069,
1.67976227728689706138237305400, 3.23156906920866213289670812530, 4.12545726433783186806020501170, 5.73237159526083871991142233325, 6.56842542247371709473487451727, 7.81739723227417578458938316570, 8.582603737128785752921989015159, 9.881516849266974991256999997949, 10.42719116453746110957907785812, 11.02508963407207415755861658178