L(s) = 1 | − 1.41·11-s − 13-s + 1.41i·17-s − 2i·19-s − 25-s − 1.41i·29-s − 2i·31-s − 1.41·47-s + 49-s − 1.41i·53-s + 1.41·59-s − 1.41·71-s − 1.41·83-s + 1.41i·101-s + 1.41i·113-s + ⋯ |
L(s) = 1 | − 1.41·11-s − 13-s + 1.41i·17-s − 2i·19-s − 25-s − 1.41i·29-s − 2i·31-s − 1.41·47-s + 49-s − 1.41i·53-s + 1.41·59-s − 1.41·71-s − 1.41·83-s + 1.41i·101-s + 1.41i·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5317435780\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5317435780\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + 1.41T + T^{2} \) |
| 17 | \( 1 - 1.41iT - T^{2} \) |
| 19 | \( 1 + 2iT - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 + 2iT - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + 1.41T + T^{2} \) |
| 53 | \( 1 + 1.41iT - T^{2} \) |
| 59 | \( 1 - 1.41T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 1.41T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + 1.41T + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.238888033339033109037856844794, −7.80326217897037996937577431535, −7.06424039525310280744158246092, −6.14195323990207397753923975105, −5.43394498138538793666014449607, −4.65401659035196692313075182019, −3.88568279264703696013386752839, −2.61746241635759860519042177421, −2.16877530925062972290686873833, −0.28014224669104248341316713065,
1.57446386799986367762378332276, 2.67584823095685293538685839265, 3.34939377710441144240597215543, 4.53082881434973993644841442036, 5.27149342614370267855257778935, 5.74070716771980778097369678175, 6.98506494550329379837474591883, 7.43031169638950712199331137177, 8.151144019914598089084978971696, 8.866454879799974382719175305811