| L(s) = 1 | − 3.02·3-s + 0.377·5-s − 0.451·7-s + 6.17·9-s + 3.14·11-s + 3.16·13-s − 1.14·15-s − 3.52·17-s − 7.48·19-s + 1.36·21-s − 2.77·23-s − 4.85·25-s − 9.62·27-s + 2.00·29-s + 7.51·31-s − 9.53·33-s − 0.170·35-s + 1.16·37-s − 9.59·39-s − 6.75·41-s − 1.46·43-s + 2.33·45-s + 8.55·47-s − 6.79·49-s + 10.6·51-s + 7.48·53-s + 1.18·55-s + ⋯ |
| L(s) = 1 | − 1.74·3-s + 0.168·5-s − 0.170·7-s + 2.05·9-s + 0.949·11-s + 0.878·13-s − 0.295·15-s − 0.855·17-s − 1.71·19-s + 0.298·21-s − 0.577·23-s − 0.971·25-s − 1.85·27-s + 0.372·29-s + 1.35·31-s − 1.66·33-s − 0.0287·35-s + 0.191·37-s − 1.53·39-s − 1.05·41-s − 0.223·43-s + 0.347·45-s + 1.24·47-s − 0.970·49-s + 1.49·51-s + 1.02·53-s + 0.160·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 3 | \( 1 + 3.02T + 3T^{2} \) |
| 5 | \( 1 - 0.377T + 5T^{2} \) |
| 7 | \( 1 + 0.451T + 7T^{2} \) |
| 11 | \( 1 - 3.14T + 11T^{2} \) |
| 13 | \( 1 - 3.16T + 13T^{2} \) |
| 17 | \( 1 + 3.52T + 17T^{2} \) |
| 19 | \( 1 + 7.48T + 19T^{2} \) |
| 23 | \( 1 + 2.77T + 23T^{2} \) |
| 29 | \( 1 - 2.00T + 29T^{2} \) |
| 31 | \( 1 - 7.51T + 31T^{2} \) |
| 37 | \( 1 - 1.16T + 37T^{2} \) |
| 41 | \( 1 + 6.75T + 41T^{2} \) |
| 43 | \( 1 + 1.46T + 43T^{2} \) |
| 47 | \( 1 - 8.55T + 47T^{2} \) |
| 53 | \( 1 - 7.48T + 53T^{2} \) |
| 59 | \( 1 + 4.90T + 59T^{2} \) |
| 61 | \( 1 - 6.00T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 - 0.732T + 71T^{2} \) |
| 73 | \( 1 - 7.15T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 - 5.84T + 83T^{2} \) |
| 89 | \( 1 + 6.91T + 89T^{2} \) |
| 97 | \( 1 - 1.47T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.172348322698512296155001937823, −6.94737769861010586024317218251, −6.34205123328177043925141849330, −6.19535420518017107095608651044, −5.21094221801011330916363794968, −4.30269800743661392942206115683, −3.88753513256716304296549054364, −2.20492181060611483900658666558, −1.17891451298771130020793790069, 0,
1.17891451298771130020793790069, 2.20492181060611483900658666558, 3.88753513256716304296549054364, 4.30269800743661392942206115683, 5.21094221801011330916363794968, 6.19535420518017107095608651044, 6.34205123328177043925141849330, 6.94737769861010586024317218251, 8.172348322698512296155001937823
