| L(s) = 1 | + 7-s + 4·11-s + 3·13-s − 7·17-s − 6·19-s + 9·23-s − 3·29-s + 7·31-s + 10·37-s − 41-s + 13·43-s − 2·47-s + 49-s − 53-s − 11·59-s − 13·61-s − 8·71-s + 8·73-s + 4·77-s − 4·79-s − 7·83-s − 14·89-s + 3·91-s + 8·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.20·11-s + 0.832·13-s − 1.69·17-s − 1.37·19-s + 1.87·23-s − 0.557·29-s + 1.25·31-s + 1.64·37-s − 0.156·41-s + 1.98·43-s − 0.291·47-s + 1/7·49-s − 0.137·53-s − 1.43·59-s − 1.66·61-s − 0.949·71-s + 0.936·73-s + 0.455·77-s − 0.450·79-s − 0.768·83-s − 1.48·89-s + 0.314·91-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.003313525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.003313525\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 13 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
| 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
−13.84903579119018, −13.09185743779306, −12.93885675968544, −12.36860812894032, −11.55446014568460, −11.33984353158112, −10.77628654003018, −10.64467049173414, −9.594187504124522, −9.203235181462115, −8.779583137866946, −8.487409000618433, −7.701486298288168, −7.184359647297946, −6.465034308349691, −6.323415970151860, −5.754655517319414, −4.762598078342479, −4.339790784189422, −4.171313563561296, −3.182783077093437, −2.646763420197229, −1.897331204513429, −1.274330745813382, −0.5768399887915460,
0.5768399887915460, 1.274330745813382, 1.897331204513429, 2.646763420197229, 3.182783077093437, 4.171313563561296, 4.339790784189422, 4.762598078342479, 5.754655517319414, 6.323415970151860, 6.465034308349691, 7.184359647297946, 7.701486298288168, 8.487409000618433, 8.779583137866946, 9.203235181462115, 9.594187504124522, 10.64467049173414, 10.77628654003018, 11.33984353158112, 11.55446014568460, 12.36860812894032, 12.93885675968544, 13.09185743779306, 13.84903579119018
