| L(s) = 1 | + 2.27·2-s + 3·3-s − 2.82·4-s − 4.54·5-s + 6.82·6-s + 7·7-s − 24.6·8-s + 9·9-s − 10.3·10-s − 40.7·11-s − 8.47·12-s + 53.2·13-s + 15.9·14-s − 13.6·15-s − 33.4·16-s + 4.54·17-s + 20.4·18-s + 122.·19-s + 12.8·20-s + 21·21-s − 92.7·22-s + 131.·23-s − 73.8·24-s − 104.·25-s + 121.·26-s + 27·27-s − 19.7·28-s + ⋯ |
| L(s) = 1 | + 0.804·2-s + 0.577·3-s − 0.353·4-s − 0.406·5-s + 0.464·6-s + 0.377·7-s − 1.08·8-s + 0.333·9-s − 0.327·10-s − 1.11·11-s − 0.203·12-s + 1.13·13-s + 0.303·14-s − 0.234·15-s − 0.522·16-s + 0.0649·17-s + 0.268·18-s + 1.48·19-s + 0.143·20-s + 0.218·21-s − 0.898·22-s + 1.19·23-s − 0.628·24-s − 0.834·25-s + 0.914·26-s + 0.192·27-s − 0.133·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.522162913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.522162913\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 - 2.27T + 8T^{2} \) |
| 5 | \( 1 + 4.54T + 125T^{2} \) |
| 11 | \( 1 + 40.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 53.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 4.54T + 4.91e3T^{2} \) |
| 19 | \( 1 - 122.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 131.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 216.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 251.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 11.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 111.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 369.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 262.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 567.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 839.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 485.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 333.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 590.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 490.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 121.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 609.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 719.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 637.T + 9.12e5T^{2} \) |
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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
−17.98754763658970438917881986449, −15.98086122027751495503143187930, −14.95781607476672701284447506651, −13.73575929130420354671972248254, −12.86151962394381771607778953262, −11.21421215607790565009657466779, −9.243114880363181102766207436345, −7.77893703428859490642609174030, −5.34722196807412904954605676407, −3.52383899701414130274827904738,
3.52383899701414130274827904738, 5.34722196807412904954605676407, 7.77893703428859490642609174030, 9.243114880363181102766207436345, 11.21421215607790565009657466779, 12.86151962394381771607778953262, 13.73575929130420354671972248254, 14.95781607476672701284447506651, 15.98086122027751495503143187930, 17.98754763658970438917881986449
