| L(s) = 1 | − 3.23·3-s − 1.23·5-s + 7-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 − 3.23·21-s + 4·23-s − 3.47·25-s − 14.4·27-s + 4.47·29-s + 6.47·31-s − 8.00·33-s − 1.23·35-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 + 49-s + 14.4·51-s − 10·53-s − 3.05·55-s + ⋯ |
| L(s) = 1 | − 1.86·3-s − 0.552·5-s + 0.377·7-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 − 0.706·21-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 − 0.208·35-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 + 0.142·49-s + 2.02·51-s − 1.37·53-s − 0.412·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7030403323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7030403323\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 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
−11.89128870713481800423031329694, −11.34032462353600206896224182854, −10.80313266572489459987418726777, −9.540661422323866214856672503338, −8.172233611398313880280940954613, −6.81721882641465152602144438016, −6.16847522482240139012874241475, −4.93937829414410636222363652171, −3.97854214530522441719437038940, −1.09406892649004172180256059289,
1.09406892649004172180256059289, 3.97854214530522441719437038940, 4.93937829414410636222363652171, 6.16847522482240139012874241475, 6.81721882641465152602144438016, 8.172233611398313880280940954613, 9.540661422323866214856672503338, 10.80313266572489459987418726777, 11.34032462353600206896224182854, 11.89128870713481800423031329694
