L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 11-s − 3·13-s + 14-s + 16-s + 8·17-s − 3·19-s − 22-s − 6·23-s − 3·26-s + 28-s − 6·29-s − 4·31-s + 32-s + 8·34-s − 2·37-s − 3·38-s − 11·41-s − 43-s − 44-s − 6·46-s − 47-s + 49-s − 3·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 0.301·11-s − 0.832·13-s + 0.267·14-s + 1/4·16-s + 1.94·17-s − 0.688·19-s − 0.213·22-s − 1.25·23-s − 0.588·26-s + 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 1.37·34-s − 0.328·37-s − 0.486·38-s − 1.71·41-s − 0.152·43-s − 0.150·44-s − 0.884·46-s − 0.145·47-s + 1/7·49-s − 0.416·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.50497819389855751237713837020, −6.59788487551126754548506984690, −5.77681567302171082156520036288, −5.33258509131763330613282269062, −4.65379605198411918849061820626, −3.77897272297251269969679810219, −3.21272968600666125911987979012, −2.19652260487469449650208200845, −1.51864247643044039261478452422, 0,
1.51864247643044039261478452422, 2.19652260487469449650208200845, 3.21272968600666125911987979012, 3.77897272297251269969679810219, 4.65379605198411918849061820626, 5.33258509131763330613282269062, 5.77681567302171082156520036288, 6.59788487551126754548506984690, 7.50497819389855751237713837020