| L(s) = 1 | − 2-s − 4-s − 5-s + 7-s + 3·8-s + 10-s + 13-s − 14-s − 16-s + 5·17-s + 2·19-s + 20-s − 6·23-s − 4·25-s − 26-s − 28-s + 29-s − 3·31-s − 5·32-s − 5·34-s − 35-s − 8·37-s − 2·38-s − 3·40-s + 2·41-s + 11·43-s + 6·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 0.377·7-s + 1.06·8-s + 0.316·10-s + 0.277·13-s − 0.267·14-s − 1/4·16-s + 1.21·17-s + 0.458·19-s + 0.223·20-s − 1.25·23-s − 4/5·25-s − 0.196·26-s − 0.188·28-s + 0.185·29-s − 0.538·31-s − 0.883·32-s − 0.857·34-s − 0.169·35-s − 1.31·37-s − 0.324·38-s − 0.474·40-s + 0.312·41-s + 1.67·43-s + 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−13.26723288540344, −12.55165353620442, −12.27491252836319, −11.82287042699709, −11.27342391347807, −10.74088569560549, −10.36811222514042, −9.860846178347861, −9.464689359995244, −8.987031510724394, −8.421426600696963, −8.037715228065255, −7.622241767018431, −7.332198701820197, −6.633290551672423, −5.772468477286451, −5.599778213535188, −4.958076657653377, −4.340963201835944, −3.752335518250052, −3.606135172007548, −2.621521059081335, −1.933673303232036, −1.328320783134166, −0.7355496663609853, 0,
0.7355496663609853, 1.328320783134166, 1.933673303232036, 2.621521059081335, 3.606135172007548, 3.752335518250052, 4.340963201835944, 4.958076657653377, 5.599778213535188, 5.772468477286451, 6.633290551672423, 7.332198701820197, 7.622241767018431, 8.037715228065255, 8.421426600696963, 8.987031510724394, 9.464689359995244, 9.860846178347861, 10.36811222514042, 10.74088569560549, 11.27342391347807, 11.82287042699709, 12.27491252836319, 12.55165353620442, 13.26723288540344
