L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s − 2·13-s − 14-s + 16-s − 6·17-s − 20-s + 25-s + 2·26-s + 28-s − 8·31-s − 32-s + 6·34-s − 35-s − 2·37-s + 40-s − 6·41-s − 10·43-s + 49-s − 50-s − 2·52-s + 6·53-s − 56-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.223·20-s + 1/5·25-s + 0.392·26-s + 0.188·28-s − 1.43·31-s − 0.176·32-s + 1.02·34-s − 0.169·35-s − 0.328·37-s + 0.158·40-s − 0.937·41-s − 1.52·43-s + 1/7·49-s − 0.141·50-s − 0.277·52-s + 0.824·53-s − 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 227430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 227430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−13.15643418920940, −12.77856523263471, −11.91499272373831, −11.74092449052834, −11.41206066473032, −10.72323430779251, −10.44787911182579, −9.944850945879462, −9.344376105691518, −8.883068787283736, −8.482401176982721, −8.155678397222716, −7.408382847061734, −7.038124154652540, −6.804014231845094, −6.041047212198634, −5.454805078406646, −4.953464071324939, −4.409682288250665, −3.832019630503310, −3.260280073704520, −2.584991736690368, −1.962077998986172, −1.592013844966698, −0.5868301578925793, 0,
0.5868301578925793, 1.592013844966698, 1.962077998986172, 2.584991736690368, 3.260280073704520, 3.832019630503310, 4.409682288250665, 4.953464071324939, 5.454805078406646, 6.041047212198634, 6.804014231845094, 7.038124154652540, 7.408382847061734, 8.155678397222716, 8.482401176982721, 8.883068787283736, 9.344376105691518, 9.944850945879462, 10.44787911182579, 10.72323430779251, 11.41206066473032, 11.74092449052834, 11.91499272373831, 12.77856523263471, 13.15643418920940