L(s) = 1 | + 5-s + 2·7-s + 11-s + 8·17-s + 8·19-s + 4·23-s + 25-s + 6·29-s + 2·35-s + 6·37-s + 2·41-s − 2·43-s − 4·47-s − 3·49-s + 2·53-s + 55-s − 12·59-s − 6·61-s − 8·67-s − 8·73-s + 2·77-s − 4·79-s − 6·83-s + 8·85-s + 10·89-s + 8·95-s − 10·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.755·7-s + 0.301·11-s + 1.94·17-s + 1.83·19-s + 0.834·23-s + 1/5·25-s + 1.11·29-s + 0.338·35-s + 0.986·37-s + 0.312·41-s − 0.304·43-s − 0.583·47-s − 3/7·49-s + 0.274·53-s + 0.134·55-s − 1.56·59-s − 0.768·61-s − 0.977·67-s − 0.936·73-s + 0.227·77-s − 0.450·79-s − 0.658·83-s + 0.867·85-s + 1.05·89-s + 0.820·95-s − 1.01·97-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
\(L(1)\) |
\(\approx\) |
\(3.231151622\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.231151622\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.70247204245896671533285986055, −7.38568704682180022940902547552, −6.37605363780843073279501720028, −5.67822318157653511731459785841, −5.10141012429750655583071883652, −4.46242359596627831443258101677, −3.27950777267665627754332466514, −2.88141264327361586576970824688, −1.48535537733964239050177571519, −1.04823756973130543281430628709,
1.04823756973130543281430628709, 1.48535537733964239050177571519, 2.88141264327361586576970824688, 3.27950777267665627754332466514, 4.46242359596627831443258101677, 5.10141012429750655583071883652, 5.67822318157653511731459785841, 6.37605363780843073279501720028, 7.38568704682180022940902547552, 7.70247204245896671533285986055