L(s) = 1 | + 2-s + 4-s − 5-s + 3·7-s + 8-s − 10-s − 11-s − 6·13-s + 3·14-s + 16-s − 6·17-s − 2·19-s − 20-s − 22-s − 8·23-s + 25-s − 6·26-s + 3·28-s + 2·29-s + 32-s − 6·34-s − 3·35-s − 2·38-s − 40-s + 6·41-s − 44-s − 8·46-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.13·7-s + 0.353·8-s − 0.316·10-s − 0.301·11-s − 1.66·13-s + 0.801·14-s + 1/4·16-s − 1.45·17-s − 0.458·19-s − 0.223·20-s − 0.213·22-s − 1.66·23-s + 1/5·25-s − 1.17·26-s + 0.566·28-s + 0.371·29-s + 0.176·32-s − 1.02·34-s − 0.507·35-s − 0.324·38-s − 0.158·40-s + 0.937·41-s − 0.150·44-s − 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 43 | \( 1 \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−13.64218945459422, −12.83663424268055, −12.58564103028407, −12.04347729743447, −11.61488930862542, −11.32037684812774, −10.68920695518428, −10.26739468645351, −9.862823509132730, −8.970307826394953, −8.712026629345145, −8.004872999622307, −7.534083332125399, −7.372246678567874, −6.563973984644300, −6.145424210216615, −5.429396193674834, −4.970994100680174, −4.460478711719636, −4.212462180583606, −3.576954312668939, −2.551472008730028, −2.315769985029897, −1.880934144821495, −0.7952881559592904, 0,
0.7952881559592904, 1.880934144821495, 2.315769985029897, 2.551472008730028, 3.576954312668939, 4.212462180583606, 4.460478711719636, 4.970994100680174, 5.429396193674834, 6.145424210216615, 6.563973984644300, 7.372246678567874, 7.534083332125399, 8.004872999622307, 8.712026629345145, 8.970307826394953, 9.862823509132730, 10.26739468645351, 10.68920695518428, 11.32037684812774, 11.61488930862542, 12.04347729743447, 12.58564103028407, 12.83663424268055, 13.64218945459422