| L(s) = 1 | − 3·3-s − 5·5-s + 24.8·7-s + 9·9-s + 25.7·11-s − 60.6·13-s + 15·15-s − 28.6·17-s − 86.6·19-s − 74.6·21-s − 52.3·23-s + 25·25-s − 27·27-s − 6·29-s + 84.8·31-s − 77.2·33-s − 124.·35-s + 448.·37-s + 181.·39-s + 183.·41-s − 252·43-s − 45·45-s + 41.9·47-s + 275.·49-s + 85.8·51-s + 228.·53-s − 128.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.34·7-s + 0.333·9-s + 0.705·11-s − 1.29·13-s + 0.258·15-s − 0.408·17-s − 1.04·19-s − 0.775·21-s − 0.474·23-s + 0.200·25-s − 0.192·27-s − 0.0384·29-s + 0.491·31-s − 0.407·33-s − 0.600·35-s + 1.99·37-s + 0.746·39-s + 0.697·41-s − 0.893·43-s − 0.149·45-s + 0.130·47-s + 0.802·49-s + 0.235·51-s + 0.592·53-s − 0.315·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 - 24.8T + 343T^{2} \) |
| 11 | \( 1 - 25.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 60.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 28.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 86.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 52.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 84.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 448.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 183.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 252T + 7.95e4T^{2} \) |
| 47 | \( 1 - 41.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 228.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 179.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 480.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 855.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 675.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 621.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 513.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.28e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.00e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 300.T + 9.12e5T^{2} \) |
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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
−9.225600166659079210595341638393, −8.253051636669725852805334405066, −7.59008732468331703818708206758, −6.69532625964738427571484050937, −5.71373009445665350309174598761, −4.55883733018664413034810037794, −4.29738465509425374665828107266, −2.53265679689631831646673920540, −1.38894698871381921938861618713, 0,
1.38894698871381921938861618713, 2.53265679689631831646673920540, 4.29738465509425374665828107266, 4.55883733018664413034810037794, 5.71373009445665350309174598761, 6.69532625964738427571484050937, 7.59008732468331703818708206758, 8.253051636669725852805334405066, 9.225600166659079210595341638393
