L(s) = 1 | + 5-s − 2·7-s − 2·13-s + 6·17-s − 23-s + 25-s − 2·35-s − 8·37-s − 12·41-s − 2·43-s + 8·47-s − 3·49-s − 10·53-s + 14·59-s + 2·61-s − 2·65-s + 2·67-s + 2·71-s − 6·73-s − 4·83-s + 6·85-s − 10·89-s + 4·91-s − 12·97-s + 14·103-s + 4·107-s + 18·109-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.755·7-s − 0.554·13-s + 1.45·17-s − 0.208·23-s + 1/5·25-s − 0.338·35-s − 1.31·37-s − 1.87·41-s − 0.304·43-s + 1.16·47-s − 3/7·49-s − 1.37·53-s + 1.82·59-s + 0.256·61-s − 0.248·65-s + 0.244·67-s + 0.237·71-s − 0.702·73-s − 0.439·83-s + 0.650·85-s − 1.05·89-s + 0.419·91-s − 1.21·97-s + 1.37·103-s + 0.386·107-s + 1.72·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−7.31281719333622242029575743903, −6.84130565934262632834471136617, −6.04546577364218806357602024148, −5.41544410890848062084277923990, −4.80661114666177355508637823999, −3.66655136224168832532270836190, −3.18917489973007361459221415350, −2.24055158761341564006731120806, −1.28123212506324439197260224916, 0,
1.28123212506324439197260224916, 2.24055158761341564006731120806, 3.18917489973007361459221415350, 3.66655136224168832532270836190, 4.80661114666177355508637823999, 5.41544410890848062084277923990, 6.04546577364218806357602024148, 6.84130565934262632834471136617, 7.31281719333622242029575743903