L(s) = 1 | + i·2-s − 4-s − 1.73·5-s − i·8-s − 1.73i·10-s − i·11-s + 16-s + 1.73·20-s + 22-s + 1.99·25-s + i·29-s + 1.73i·31-s + i·32-s + 1.73i·40-s + i·44-s + ⋯ |
L(s) = 1 | + i·2-s − 4-s − 1.73·5-s − i·8-s − 1.73i·10-s − i·11-s + 16-s + 1.73·20-s + 22-s + 1.99·25-s + i·29-s + 1.73i·31-s + i·32-s + 1.73i·40-s + i·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5717469621\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5717469621\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 1.73T + T^{2} \) |
| 11 | \( 1 + iT - T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - iT - T^{2} \) |
| 31 | \( 1 - 1.73iT - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - iT - T^{2} \) |
| 59 | \( 1 + 1.73T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( 1 - 1.73T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - 1.73iT - T^{2} \) |
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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
−8.734622149813180270690000644312, −8.173216074995425293347818336289, −7.58122414654573459107572826813, −6.92408754919693638213430329843, −6.19977681486935614640579465977, −5.16942615048672197712307958247, −4.56781568875849452329169665779, −3.61052633163461439627682900120, −3.18061693015274264072762620974, −0.993200744814707287351114541534,
0.44221936371038336493023303586, 1.93914934136199384104800308841, 2.96408638745224397027568615899, 3.87728776829330413822772124634, 4.34073256674451001629812577321, 5.03903684800058156416594910986, 6.22697295273589252441928874691, 7.36478488934704653931739490302, 7.82647678313390958979881922868, 8.438698510785410031078001938741