L(s) = 1 | − 2-s − 4-s − 4·7-s + 3·8-s + 2·13-s + 4·14-s − 16-s + 17-s − 4·19-s + 4·23-s − 2·26-s + 4·28-s − 6·29-s + 4·31-s − 5·32-s − 34-s + 2·37-s + 4·38-s + 6·41-s − 4·43-s − 4·46-s + 9·49-s − 2·52-s + 6·53-s − 12·56-s + 6·58-s + 12·59-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 1.51·7-s + 1.06·8-s + 0.554·13-s + 1.06·14-s − 1/4·16-s + 0.242·17-s − 0.917·19-s + 0.834·23-s − 0.392·26-s + 0.755·28-s − 1.11·29-s + 0.718·31-s − 0.883·32-s − 0.171·34-s + 0.328·37-s + 0.648·38-s + 0.937·41-s − 0.609·43-s − 0.589·46-s + 9/7·49-s − 0.277·52-s + 0.824·53-s − 1.60·56-s + 0.787·58-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.290892360612030079383566382264, −7.47752919679876584865535315921, −6.71231760010847881949462194838, −6.04937616148002008106105333421, −5.15552129752651214519944450440, −4.12175372865283449997475235565, −3.51310931579236422842332858140, −2.45057549626766878513681501691, −1.08638100125964830130657853489, 0,
1.08638100125964830130657853489, 2.45057549626766878513681501691, 3.51310931579236422842332858140, 4.12175372865283449997475235565, 5.15552129752651214519944450440, 6.04937616148002008106105333421, 6.71231760010847881949462194838, 7.47752919679876584865535315921, 8.290892360612030079383566382264