L(s) = 1 | + (0.338 − 1.04i)2-s + (−0.809 + 0.587i)3-s + (0.647 + 0.470i)4-s + (0.309 + 0.951i)5-s + (0.338 + 1.04i)6-s + (0.570 + 0.414i)7-s + (2.48 − 1.80i)8-s + (0.309 − 0.951i)9-s + 1.09·10-s + (3.31 − 0.189i)11-s − 0.800·12-s + (−1.45 + 4.48i)13-s + (0.624 − 0.453i)14-s + (−0.809 − 0.587i)15-s + (−0.543 − 1.67i)16-s + (−2.40 − 7.39i)17-s + ⋯ |
L(s) = 1 | + (0.239 − 0.736i)2-s + (−0.467 + 0.339i)3-s + (0.323 + 0.235i)4-s + (0.138 + 0.425i)5-s + (0.138 + 0.425i)6-s + (0.215 + 0.156i)7-s + (0.877 − 0.637i)8-s + (0.103 − 0.317i)9-s + 0.346·10-s + (0.998 − 0.0572i)11-s − 0.231·12-s + (−0.403 + 1.24i)13-s + (0.166 − 0.121i)14-s + (−0.208 − 0.151i)15-s + (−0.135 − 0.418i)16-s + (−0.583 − 1.79i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 + 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.32265 - 0.140135i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.32265 - 0.140135i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.809 - 0.587i)T \) |
| 5 | \( 1 + (-0.309 - 0.951i)T \) |
| 11 | \( 1 + (-3.31 + 0.189i)T \) |
good | 2 | \( 1 + (-0.338 + 1.04i)T + (-1.61 - 1.17i)T^{2} \) |
| 7 | \( 1 + (-0.570 - 0.414i)T + (2.16 + 6.65i)T^{2} \) |
| 13 | \( 1 + (1.45 - 4.48i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (2.40 + 7.39i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.970 + 0.705i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + 6.89T + 23T^{2} \) |
| 29 | \( 1 + (1.07 + 0.780i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (2.37 - 7.30i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (6.82 + 4.95i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-0.188 + 0.136i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 7.32T + 43T^{2} \) |
| 47 | \( 1 + (6.73 - 4.89i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-2.10 + 6.48i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-2.86 - 2.08i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (3.35 + 10.3i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + 2.04T + 67T^{2} \) |
| 71 | \( 1 + (-0.207 - 0.637i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-4.04 - 2.94i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-0.704 + 2.16i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (0.652 + 2.00i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + 3.34T + 89T^{2} \) |
| 97 | \( 1 + (-1.02 + 3.15i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−12.36281627830851208039281795512, −11.66122025344899999512966909663, −11.17799077752916944502710654392, −9.955900443099076271472792763137, −9.081693572046099048948033338899, −7.27909519562743790691105591392, −6.53183999294914563657652342135, −4.82705091919757726414983975721, −3.65582911305122491411455168921, −2.07205358455758633746043484407,
1.72302966651076402732363805623, 4.23527958756584545564956852580, 5.63387844351742828677188915659, 6.27156984678481161726056086168, 7.52533138284867270678634344165, 8.396907083547202467099791722201, 10.01212450397809253758298639440, 10.88726561297810152580408215730, 11.97695544591643191301203214131, 12.89854451581403270878909505186