| L(s) = 1 | + 2·7-s − 4·11-s − 4·13-s + 4·19-s + 2·23-s − 2·29-s − 4·37-s − 2·41-s − 6·43-s + 6·47-s − 3·49-s − 4·53-s − 12·59-s − 10·61-s + 14·67-s + 8·71-s − 8·73-s − 8·77-s − 16·79-s − 2·83-s − 6·89-s − 8·91-s − 16·97-s − 6·101-s + 14·103-s + 10·107-s − 6·109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1.20·11-s − 1.10·13-s + 0.917·19-s + 0.417·23-s − 0.371·29-s − 0.657·37-s − 0.312·41-s − 0.914·43-s + 0.875·47-s − 3/7·49-s − 0.549·53-s − 1.56·59-s − 1.28·61-s + 1.71·67-s + 0.949·71-s − 0.936·73-s − 0.911·77-s − 1.80·79-s − 0.219·83-s − 0.635·89-s − 0.838·91-s − 1.62·97-s − 0.597·101-s + 1.37·103-s + 0.966·107-s − 0.574·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.030554265249448297712609550478, −7.54734161635556487268348713556, −6.89500255860999856880586417200, −5.72561147857810855234807736185, −5.08928054572465165404463903083, −4.59174506533192788492533024086, −3.32230357018259482500851733436, −2.53583091780922087841565970435, −1.51948873935140986368359744751, 0,
1.51948873935140986368359744751, 2.53583091780922087841565970435, 3.32230357018259482500851733436, 4.59174506533192788492533024086, 5.08928054572465165404463903083, 5.72561147857810855234807736185, 6.89500255860999856880586417200, 7.54734161635556487268348713556, 8.030554265249448297712609550478
