| L(s) = 1 | + 24.5·3-s + 25·5-s + 49·7-s + 359.·9-s + 90.2·11-s − 14.4·13-s + 613.·15-s + 407.·17-s + 2.28e3·19-s + 1.20e3·21-s − 505.·23-s + 625·25-s + 2.85e3·27-s − 3.16e3·29-s − 6.23e3·31-s + 2.21e3·33-s + 1.22e3·35-s + 5.38e3·37-s − 354.·39-s + 1.17e4·41-s − 5.82e3·43-s + 8.98e3·45-s + 7.34e3·47-s + 2.40e3·49-s + 9.99e3·51-s + 1.49e4·53-s + 2.25e3·55-s + ⋯ |
| L(s) = 1 | + 1.57·3-s + 0.447·5-s + 0.377·7-s + 1.47·9-s + 0.224·11-s − 0.0236·13-s + 0.704·15-s + 0.341·17-s + 1.45·19-s + 0.595·21-s − 0.199·23-s + 0.200·25-s + 0.754·27-s − 0.698·29-s − 1.16·31-s + 0.354·33-s + 0.169·35-s + 0.647·37-s − 0.0373·39-s + 1.09·41-s − 0.480·43-s + 0.661·45-s + 0.485·47-s + 0.142·49-s + 0.538·51-s + 0.732·53-s + 0.100·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.836120161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.836120161\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 7 | \( 1 - 49T \) |
| good | 3 | \( 1 - 24.5T + 243T^{2} \) |
| 11 | \( 1 - 90.2T + 1.61e5T^{2} \) |
| 13 | \( 1 + 14.4T + 3.71e5T^{2} \) |
| 17 | \( 1 - 407.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.28e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 505.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.16e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.23e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.38e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.17e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.82e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 7.34e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.49e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.71e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.28e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.88e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.85e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.59e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.91e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.14e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.71e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.15e4T + 8.58e9T^{2} \) |
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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
−12.49108653150524367768195427712, −11.15155187515066393564973953431, −9.757465862979467932520795011601, −9.180044296371370195865357031808, −8.054727619267592753293839216260, −7.22511552429934842807447086281, −5.52203918705780333620702924555, −3.92346835388940290120785466087, −2.74191624921347002391444778744, −1.47287247417362097021632997009,
1.47287247417362097021632997009, 2.74191624921347002391444778744, 3.92346835388940290120785466087, 5.52203918705780333620702924555, 7.22511552429934842807447086281, 8.054727619267592753293839216260, 9.180044296371370195865357031808, 9.757465862979467932520795011601, 11.15155187515066393564973953431, 12.49108653150524367768195427712
