| L(s) = 1 | + 19.9·3-s − 106.·5-s + 49·7-s + 156.·9-s − 452.·11-s − 886.·13-s − 2.12e3·15-s + 297.·17-s + 2.28e3·19-s + 978.·21-s − 555.·23-s + 8.14e3·25-s − 1.73e3·27-s + 8.26e3·29-s − 4.24e3·31-s − 9.03e3·33-s − 5.20e3·35-s + 758.·37-s − 1.77e4·39-s + 1.72e4·41-s − 5.37e3·43-s − 1.65e4·45-s + 2.56e4·47-s + 2.40e3·49-s + 5.95e3·51-s − 1.08e4·53-s + 4.80e4·55-s + ⋯ |
| L(s) = 1 | + 1.28·3-s − 1.89·5-s + 0.377·7-s + 0.642·9-s − 1.12·11-s − 1.45·13-s − 2.43·15-s + 0.250·17-s + 1.45·19-s + 0.484·21-s − 0.218·23-s + 2.60·25-s − 0.458·27-s + 1.82·29-s − 0.793·31-s − 1.44·33-s − 0.717·35-s + 0.0911·37-s − 1.86·39-s + 1.59·41-s − 0.443·43-s − 1.22·45-s + 1.69·47-s + 0.142·49-s + 0.320·51-s − 0.529·53-s + 2.14·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.817978884\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.817978884\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 49T \) |
| good | 3 | \( 1 - 19.9T + 243T^{2} \) |
| 5 | \( 1 + 106.T + 3.12e3T^{2} \) |
| 11 | \( 1 + 452.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 886.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 297.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.28e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 555.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.26e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.24e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 758.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.72e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.37e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.56e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.08e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.79e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 8.46e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.43e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.11e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.00e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.53e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.43e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.03e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.30e4T + 8.58e9T^{2} \) |
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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
−10.23096957490264221250857643221, −9.184411434767617985481926968496, −8.205292313340335857147824730231, −7.68265108143760142150628989524, −7.26635396881615425826118859107, −5.19871357165099204775188754814, −4.30396408458670992657539385461, −3.21075401051951307698509459443, −2.54130234799171958191716627889, −0.63242666027610530723386839012,
0.63242666027610530723386839012, 2.54130234799171958191716627889, 3.21075401051951307698509459443, 4.30396408458670992657539385461, 5.19871357165099204775188754814, 7.26635396881615425826118859107, 7.68265108143760142150628989524, 8.205292313340335857147824730231, 9.184411434767617985481926968496, 10.23096957490264221250857643221
