| L(s) = 1 | + 2·7-s + 2·11-s + 2·13-s − 6·17-s − 8·19-s − 4·23-s − 8·29-s − 10·37-s − 2·41-s + 12·43-s − 3·49-s − 10·53-s − 6·59-s + 2·61-s + 8·67-s − 4·71-s + 4·73-s + 4·77-s + 8·79-s + 4·83-s − 6·89-s + 4·91-s + 8·97-s + 2·103-s + 4·107-s − 6·109-s − 2·113-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.603·11-s + 0.554·13-s − 1.45·17-s − 1.83·19-s − 0.834·23-s − 1.48·29-s − 1.64·37-s − 0.312·41-s + 1.82·43-s − 3/7·49-s − 1.37·53-s − 0.781·59-s + 0.256·61-s + 0.977·67-s − 0.474·71-s + 0.468·73-s + 0.455·77-s + 0.900·79-s + 0.439·83-s − 0.635·89-s + 0.419·91-s + 0.812·97-s + 0.197·103-s + 0.386·107-s − 0.574·109-s − 0.188·113-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 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.304131594473762327756515411723, −7.49164756417775038761367556330, −6.56131643384857882726206550272, −6.11711230688273026826549465002, −5.05680823533136410317774645046, −4.25297727101963485280140324098, −3.70555749611452981961247982149, −2.24693818452534816487808793489, −1.66209771157902873211308741476, 0,
1.66209771157902873211308741476, 2.24693818452534816487808793489, 3.70555749611452981961247982149, 4.25297727101963485280140324098, 5.05680823533136410317774645046, 6.11711230688273026826549465002, 6.56131643384857882726206550272, 7.49164756417775038761367556330, 8.304131594473762327756515411723
