L(s) = 1 | + 5-s − 2·7-s − 4·11-s − 2·13-s + 5·17-s + 5·19-s − 23-s + 25-s − 2·29-s − 7·31-s − 2·35-s − 6·37-s − 4·43-s − 4·47-s − 3·49-s + 9·53-s − 4·55-s − 14·59-s − 11·61-s − 2·65-s − 14·67-s − 12·73-s + 8·77-s + 3·79-s + 83-s + 5·85-s + 4·91-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.755·7-s − 1.20·11-s − 0.554·13-s + 1.21·17-s + 1.14·19-s − 0.208·23-s + 1/5·25-s − 0.371·29-s − 1.25·31-s − 0.338·35-s − 0.986·37-s − 0.609·43-s − 0.583·47-s − 3/7·49-s + 1.23·53-s − 0.539·55-s − 1.82·59-s − 1.40·61-s − 0.248·65-s − 1.71·67-s − 1.40·73-s + 0.911·77-s + 0.337·79-s + 0.109·83-s + 0.542·85-s + 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−8.823447724723373773807392609659, −7.61860649519705384132807251020, −7.40844736397527203482945634048, −6.19968378010274074206707025068, −5.49848998578718543925454313897, −4.88282976724323283759106636909, −3.45319542610164611802228537966, −2.89101337098531179638687733013, −1.63994653615646372792011785412, 0,
1.63994653615646372792011785412, 2.89101337098531179638687733013, 3.45319542610164611802228537966, 4.88282976724323283759106636909, 5.49848998578718543925454313897, 6.19968378010274074206707025068, 7.40844736397527203482945634048, 7.61860649519705384132807251020, 8.823447724723373773807392609659