| L(s) = 1 | − 2-s + 4-s − 8-s − 5·11-s + 13-s + 16-s − 3·17-s − 5·19-s + 5·22-s + 9·23-s − 26-s + 29-s + 2·31-s − 32-s + 3·34-s − 3·37-s + 5·38-s − 12·41-s − 7·43-s − 5·44-s − 9·46-s + 4·47-s + 52-s − 10·53-s − 58-s + 14·59-s + 11·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.50·11-s + 0.277·13-s + 1/4·16-s − 0.727·17-s − 1.14·19-s + 1.06·22-s + 1.87·23-s − 0.196·26-s + 0.185·29-s + 0.359·31-s − 0.176·32-s + 0.514·34-s − 0.493·37-s + 0.811·38-s − 1.87·41-s − 1.06·43-s − 0.753·44-s − 1.32·46-s + 0.583·47-s + 0.138·52-s − 1.37·53-s − 0.131·58-s + 1.82·59-s + 1.40·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7493681908\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7493681908\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.78926302561804, −12.31375372092339, −11.57883720580382, −11.29308250797647, −10.80830043761103, −10.39969016176136, −10.08293748134931, −9.558469260612149, −8.824243062383759, −8.477781140352046, −8.394472951425704, −7.587854469242304, −7.172612277363423, −6.636113247709244, −6.380826740226251, −5.574376844912344, −5.006795493865300, −4.855247845609055, −3.991491394058441, −3.325151993693559, −2.875414069323927, −2.261010068954915, −1.835158439883726, −0.9999882331163736, −0.2877209013079685,
0.2877209013079685, 0.9999882331163736, 1.835158439883726, 2.261010068954915, 2.875414069323927, 3.325151993693559, 3.991491394058441, 4.855247845609055, 5.006795493865300, 5.574376844912344, 6.380826740226251, 6.636113247709244, 7.172612277363423, 7.587854469242304, 8.394472951425704, 8.477781140352046, 8.824243062383759, 9.558469260612149, 10.08293748134931, 10.39969016176136, 10.80830043761103, 11.29308250797647, 11.57883720580382, 12.31375372092339, 12.78926302561804
