L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 3·9-s − 2·11-s − 2·13-s + 14-s + 16-s + 2·17-s + 3·18-s − 6·19-s + 2·22-s − 5·25-s + 2·26-s − 28-s + 8·29-s − 31-s − 32-s − 2·34-s − 3·36-s − 8·37-s + 6·38-s − 10·41-s − 6·43-s − 2·44-s − 4·47-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 9-s − 0.603·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.707·18-s − 1.37·19-s + 0.426·22-s − 25-s + 0.392·26-s − 0.188·28-s + 1.48·29-s − 0.179·31-s − 0.176·32-s − 0.342·34-s − 1/2·36-s − 1.31·37-s + 0.973·38-s − 1.56·41-s − 0.914·43-s − 0.301·44-s − 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 434 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 434 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 31 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−10.42398618178035066302830692228, −9.966181263423740547879905328438, −8.692103692037522457493629664511, −8.222505697305419609916947180227, −7.03583909323380574684214200002, −6.10142362743840668568183330623, −5.02689386857091064180337724920, −3.37307842458307700498184529678, −2.20827273495555998462564988342, 0,
2.20827273495555998462564988342, 3.37307842458307700498184529678, 5.02689386857091064180337724920, 6.10142362743840668568183330623, 7.03583909323380574684214200002, 8.222505697305419609916947180227, 8.692103692037522457493629664511, 9.966181263423740547879905328438, 10.42398618178035066302830692228