L(s) = 1 | − 2·5-s − 2·7-s + 4·11-s + 2·13-s − 4·17-s + 4·19-s − 8·23-s − 25-s − 6·29-s + 6·31-s + 4·35-s − 2·37-s − 12·41-s − 12·43-s − 8·47-s − 3·49-s − 6·53-s − 8·55-s + 8·59-s − 10·61-s − 4·65-s + 8·67-s + 2·73-s − 8·77-s + 14·79-s + 12·83-s + 8·85-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 0.755·7-s + 1.20·11-s + 0.554·13-s − 0.970·17-s + 0.917·19-s − 1.66·23-s − 1/5·25-s − 1.11·29-s + 1.07·31-s + 0.676·35-s − 0.328·37-s − 1.87·41-s − 1.82·43-s − 1.16·47-s − 3/7·49-s − 0.824·53-s − 1.07·55-s + 1.04·59-s − 1.28·61-s − 0.496·65-s + 0.977·67-s + 0.234·73-s − 0.911·77-s + 1.57·79-s + 1.31·83-s + 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−9.468657214301905304141993566775, −8.492984870607399512730308262292, −7.85364633298304614829141363943, −6.68490418072117099061712878414, −6.33298440987407835664888781952, −4.98237762686229313120902385252, −3.84055323138271859535887586625, −3.42671764173860393498603657551, −1.74260187134927495442557965172, 0,
1.74260187134927495442557965172, 3.42671764173860393498603657551, 3.84055323138271859535887586625, 4.98237762686229313120902385252, 6.33298440987407835664888781952, 6.68490418072117099061712878414, 7.85364633298304614829141363943, 8.492984870607399512730308262292, 9.468657214301905304141993566775