L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 3·11-s + 13-s − 14-s + 16-s − 17-s − 2·19-s − 3·22-s + 7·23-s − 26-s + 28-s − 5·29-s − 5·31-s − 32-s + 34-s − 2·37-s + 2·38-s + 6·41-s + 5·43-s + 3·44-s − 7·46-s + 47-s + 49-s + 52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 0.904·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.458·19-s − 0.639·22-s + 1.45·23-s − 0.196·26-s + 0.188·28-s − 0.928·29-s − 0.898·31-s − 0.176·32-s + 0.171·34-s − 0.328·37-s + 0.324·38-s + 0.937·41-s + 0.762·43-s + 0.452·44-s − 1.03·46-s + 0.145·47-s + 1/7·49-s + 0.138·52-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
\(L(1)\) |
\(\approx\) |
\(1.653439480\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.653439480\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−7.54578755097313007589418808368, −7.25666578319399457934692928522, −6.41544629778896080141112298488, −5.81717779818878352994951033829, −4.97803353121862181541316230980, −4.11793125868836684390176582503, −3.42173822264390013248036651430, −2.40702484442373582559285833800, −1.60080545827127354973668615291, −0.71625308322215230191455104280,
0.71625308322215230191455104280, 1.60080545827127354973668615291, 2.40702484442373582559285833800, 3.42173822264390013248036651430, 4.11793125868836684390176582503, 4.97803353121862181541316230980, 5.81717779818878352994951033829, 6.41544629778896080141112298488, 7.25666578319399457934692928522, 7.54578755097313007589418808368