| L(s) = 1 | − 5-s + 2·7-s + 3.46·11-s − 2.73·13-s + 3.46·17-s + 19-s + 3.46·23-s + 25-s − 3.46·29-s − 1.46·31-s − 2·35-s + 6.73·37-s + 6·41-s − 4.92·43-s − 12.9·47-s − 3·49-s + 10.7·53-s − 3.46·55-s − 6.92·59-s + 12.3·61-s + 2.73·65-s + 6.73·67-s + 2.53·71-s − 0.535·73-s + 6.92·77-s + 2.92·79-s − 3.46·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s + 1.04·11-s − 0.757·13-s + 0.840·17-s + 0.229·19-s + 0.722·23-s + 0.200·25-s − 0.643·29-s − 0.262·31-s − 0.338·35-s + 1.10·37-s + 0.937·41-s − 0.751·43-s − 1.88·47-s − 0.428·49-s + 1.47·53-s − 0.467·55-s − 0.901·59-s + 1.58·61-s + 0.338·65-s + 0.822·67-s + 0.300·71-s − 0.0627·73-s + 0.789·77-s + 0.329·79-s − 0.380·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.020091570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.020091570\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 2.73T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 - 6.73T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4.92T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 - 6.73T + 67T^{2} \) |
| 71 | \( 1 - 2.53T + 71T^{2} \) |
| 73 | \( 1 + 0.535T + 73T^{2} \) |
| 79 | \( 1 - 2.92T + 79T^{2} \) |
| 83 | \( 1 + 3.46T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
| 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
−8.500944837848202769823797012696, −7.84374712086266629852550336834, −7.22070268779421723856344852706, −6.45814076891746369125045542479, −5.44835884911990703875667087530, −4.78589756435213296197656594597, −3.95291295420736884426522019352, −3.12292382116067265081520818531, −1.93274918210383358514914734067, −0.880701398752982840553281107725,
0.880701398752982840553281107725, 1.93274918210383358514914734067, 3.12292382116067265081520818531, 3.95291295420736884426522019352, 4.78589756435213296197656594597, 5.44835884911990703875667087530, 6.45814076891746369125045542479, 7.22070268779421723856344852706, 7.84374712086266629852550336834, 8.500944837848202769823797012696
