| L(s) = 1 | − 3-s + 3.22·7-s + 9-s + 2.15·11-s + 3.08·13-s − 5.11·17-s − 1.80·19-s − 3.22·21-s + 23-s − 27-s + 7.18·29-s − 1.50·31-s − 2.15·33-s + 11.1·37-s − 3.08·39-s − 6.38·41-s + 5.16·43-s + 1.19·47-s + 3.42·49-s + 5.11·51-s + 2.57·53-s + 1.80·57-s + 9.34·59-s − 7.96·61-s + 3.22·63-s + 5.30·67-s − 69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.22·7-s + 0.333·9-s + 0.650·11-s + 0.855·13-s − 1.24·17-s − 0.414·19-s − 0.704·21-s + 0.208·23-s − 0.192·27-s + 1.33·29-s − 0.270·31-s − 0.375·33-s + 1.82·37-s − 0.493·39-s − 0.997·41-s + 0.788·43-s + 0.174·47-s + 0.489·49-s + 0.716·51-s + 0.353·53-s + 0.239·57-s + 1.21·59-s − 1.01·61-s + 0.406·63-s + 0.647·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.131622777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.131622777\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 3.22T + 7T^{2} \) |
| 11 | \( 1 - 2.15T + 11T^{2} \) |
| 13 | \( 1 - 3.08T + 13T^{2} \) |
| 17 | \( 1 + 5.11T + 17T^{2} \) |
| 19 | \( 1 + 1.80T + 19T^{2} \) |
| 29 | \( 1 - 7.18T + 29T^{2} \) |
| 31 | \( 1 + 1.50T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 + 6.38T + 41T^{2} \) |
| 43 | \( 1 - 5.16T + 43T^{2} \) |
| 47 | \( 1 - 1.19T + 47T^{2} \) |
| 53 | \( 1 - 2.57T + 53T^{2} \) |
| 59 | \( 1 - 9.34T + 59T^{2} \) |
| 61 | \( 1 + 7.96T + 61T^{2} \) |
| 67 | \( 1 - 5.30T + 67T^{2} \) |
| 71 | \( 1 + 0.966T + 71T^{2} \) |
| 73 | \( 1 + 4.23T + 73T^{2} \) |
| 79 | \( 1 - 7.53T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 - 9.81T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−8.039400635204653525499898139995, −7.17752726038308593510059132012, −6.46241051567561200385367762722, −5.97201921830607990993816032872, −4.99889069400459149897301236964, −4.45858998744256999030289418772, −3.85016682611547077538067898330, −2.58006218769520747921618573152, −1.66280065390177574595571816415, −0.819828313038900899529601382802,
0.819828313038900899529601382802, 1.66280065390177574595571816415, 2.58006218769520747921618573152, 3.85016682611547077538067898330, 4.45858998744256999030289418772, 4.99889069400459149897301236964, 5.97201921830607990993816032872, 6.46241051567561200385367762722, 7.17752726038308593510059132012, 8.039400635204653525499898139995
