| L(s) = 1 | + 3.23·3-s + 1.23·5-s + 7.47·9-s + 2.47·11-s − 5.23·13-s + 4.00·15-s + 4.47·17-s − 3.23·19-s + 4·23-s − 3.47·25-s + 14.4·27-s + 4.47·29-s − 6.47·31-s + 8.00·33-s + 4.47·37-s − 16.9·39-s − 0.472·41-s − 2.47·43-s + 9.23·45-s − 1.52·47-s + 14.4·51-s − 10·53-s + 3.05·55-s − 10.4·57-s + 4.76·59-s − 6.76·61-s − 6.47·65-s + ⋯ |
| L(s) = 1 | + 1.86·3-s + 0.552·5-s + 2.49·9-s + 0.745·11-s − 1.45·13-s + 1.03·15-s + 1.08·17-s − 0.742·19-s + 0.834·23-s − 0.694·25-s + 2.78·27-s + 0.830·29-s − 1.16·31-s + 1.39·33-s + 0.735·37-s − 2.71·39-s − 0.0737·41-s − 0.376·43-s + 1.37·45-s − 0.222·47-s + 2.02·51-s − 1.37·53-s + 0.412·55-s − 1.38·57-s + 0.620·59-s − 0.866·61-s − 0.802·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.733043214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.733043214\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 3.23T + 3T^{2} \) |
| 5 | \( 1 - 1.23T + 5T^{2} \) |
| 11 | \( 1 - 2.47T + 11T^{2} \) |
| 13 | \( 1 + 5.23T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 + 3.23T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 6.47T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 + 0.472T + 41T^{2} \) |
| 43 | \( 1 + 2.47T + 43T^{2} \) |
| 47 | \( 1 + 1.52T + 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 - 4.76T + 59T^{2} \) |
| 61 | \( 1 + 6.76T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 + 4.94T + 79T^{2} \) |
| 83 | \( 1 - 4.76T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 3.52T + 97T^{2} \) |
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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
−9.529362744689139126816148281944, −8.705412889654435934657014059545, −7.889147343207897363008561799069, −7.28831557806710236450435715403, −6.42839971483428242832740500367, −5.11076632810424960799315498835, −4.16898783984654253171752289648, −3.23584617740481828152951930413, −2.41762720435726300677545954570, −1.51418020789488514632355481407,
1.51418020789488514632355481407, 2.41762720435726300677545954570, 3.23584617740481828152951930413, 4.16898783984654253171752289648, 5.11076632810424960799315498835, 6.42839971483428242832740500367, 7.28831557806710236450435715403, 7.889147343207897363008561799069, 8.705412889654435934657014059545, 9.529362744689139126816148281944
