L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 3·9-s − 11-s + 13-s + 14-s + 16-s − 6·17-s − 3·18-s + 19-s − 22-s − 23-s + 26-s + 28-s − 5·29-s − 7·31-s + 32-s − 6·34-s − 3·36-s − 8·37-s + 38-s − 3·43-s − 44-s − 46-s + 49-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 9-s − 0.301·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.707·18-s + 0.229·19-s − 0.213·22-s − 0.208·23-s + 0.196·26-s + 0.188·28-s − 0.928·29-s − 1.25·31-s + 0.176·32-s − 1.02·34-s − 1/2·36-s − 1.31·37-s + 0.162·38-s − 0.457·43-s − 0.150·44-s − 0.147·46-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 13 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−8.129684674928074132887415856409, −7.24438733346919315547207687187, −6.59706951968264100129800894758, −5.64206729101921736035290723344, −5.26484265294212836427523141356, −4.26623744249286454463451402799, −3.51548792635493529717271316644, −2.55420145820780508200827362202, −1.74482644530745088906919023146, 0,
1.74482644530745088906919023146, 2.55420145820780508200827362202, 3.51548792635493529717271316644, 4.26623744249286454463451402799, 5.26484265294212836427523141356, 5.64206729101921736035290723344, 6.59706951968264100129800894758, 7.24438733346919315547207687187, 8.129684674928074132887415856409