| 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
| \(L(1)\) |
\(\approx\) |
\(1.886282611\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.886282611\) |
| \(L(\frac{3}{2})\) |
|
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
−12.69184392290246865848785566473, −11.14940950384920267977657764595, −10.13066451433255203489171863316, −9.042883762049257040898170081217, −8.356816000659126457280838882807, −7.62017593248943357078401779319, −6.36402485764266418077390521555, −4.29668658677344589822753768806, −3.45664727824938054196263640486, −2.15948301807031587246528686001,
2.15948301807031587246528686001, 3.45664727824938054196263640486, 4.29668658677344589822753768806, 6.36402485764266418077390521555, 7.62017593248943357078401779319, 8.356816000659126457280838882807, 9.042883762049257040898170081217, 10.13066451433255203489171863316, 11.14940950384920267977657764595, 12.69184392290246865848785566473
