| L(s) = 1 | − 3.53·2-s + 3·3-s + 4.46·4-s + 5·5-s − 10.5·6-s − 7·7-s + 12.4·8-s + 9·9-s − 17.6·10-s − 2.93·11-s + 13.4·12-s − 19.0·13-s + 24.7·14-s + 15·15-s − 79.7·16-s + 122.·17-s − 31.7·18-s + 107.·19-s + 22.3·20-s − 21·21-s + 10.3·22-s + 210.·23-s + 37.4·24-s + 25·25-s + 67.3·26-s + 27·27-s − 31.2·28-s + ⋯ |
| L(s) = 1 | − 1.24·2-s + 0.577·3-s + 0.558·4-s + 0.447·5-s − 0.720·6-s − 0.377·7-s + 0.551·8-s + 0.333·9-s − 0.558·10-s − 0.0805·11-s + 0.322·12-s − 0.406·13-s + 0.471·14-s + 0.258·15-s − 1.24·16-s + 1.74·17-s − 0.416·18-s + 1.29·19-s + 0.249·20-s − 0.218·21-s + 0.100·22-s + 1.90·23-s + 0.318·24-s + 0.200·25-s + 0.507·26-s + 0.192·27-s − 0.211·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.064514029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.064514029\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 3.53T + 8T^{2} \) |
| 11 | \( 1 + 2.93T + 1.33e3T^{2} \) |
| 13 | \( 1 + 19.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 122.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 210.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 95.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 94.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 97.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 491.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 43.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 473.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 183.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 760.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 198.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 309.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 665.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 621.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 24.7T + 4.93e5T^{2} \) |
| 83 | \( 1 + 406.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 261.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.00e3T + 9.12e5T^{2} \) |
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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
−13.40546592608703245871921898326, −12.18250555097709804072936119577, −10.67678270051248069312468546548, −9.750487568024786448080816428638, −9.153033322238177666395213432806, −7.920982822273216246053602816816, −7.02876736555687686837865597329, −5.17693363736470717283113350431, −3.08354544168682833867216834671, −1.18419157274904927371218082605,
1.18419157274904927371218082605, 3.08354544168682833867216834671, 5.17693363736470717283113350431, 7.02876736555687686837865597329, 7.920982822273216246053602816816, 9.153033322238177666395213432806, 9.750487568024786448080816428638, 10.67678270051248069312468546548, 12.18250555097709804072936119577, 13.40546592608703245871921898326
