| L(s) = 1 | + 5-s − 2·7-s + 11-s + 2·13-s − 2·19-s + 25-s − 8·31-s − 2·35-s + 2·37-s − 2·43-s − 3·49-s − 6·53-s + 55-s − 12·59-s + 2·61-s + 2·65-s + 4·67-s + 2·73-s − 2·77-s + 10·79-s − 12·83-s + 6·89-s − 4·91-s − 2·95-s + 14·97-s + 4·103-s − 12·107-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s + 0.301·11-s + 0.554·13-s − 0.458·19-s + 1/5·25-s − 1.43·31-s − 0.338·35-s + 0.328·37-s − 0.304·43-s − 3/7·49-s − 0.824·53-s + 0.134·55-s − 1.56·59-s + 0.256·61-s + 0.248·65-s + 0.488·67-s + 0.234·73-s − 0.227·77-s + 1.12·79-s − 1.31·83-s + 0.635·89-s − 0.419·91-s − 0.205·95-s + 1.42·97-s + 0.394·103-s − 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.45950523429408724078642817430, −6.62809575887132393562329754634, −6.23014779417671950131080268515, −5.51350916544264899972205145523, −4.68675354902407805865003313060, −3.78604634402167547094494153595, −3.19237694755342407493957762628, −2.20902138146244912815630447797, −1.32116731583023775044904348023, 0,
1.32116731583023775044904348023, 2.20902138146244912815630447797, 3.19237694755342407493957762628, 3.78604634402167547094494153595, 4.68675354902407805865003313060, 5.51350916544264899972205145523, 6.23014779417671950131080268515, 6.62809575887132393562329754634, 7.45950523429408724078642817430
