| L(s) = 1 | + 5.79·3-s + 16.8·5-s − 7·7-s + 6.52·9-s − 11·11-s + 40.8·13-s + 97.5·15-s + 127.·17-s + 21.3·19-s − 40.5·21-s + 50.0·23-s + 158.·25-s − 118.·27-s − 233.·29-s + 91.5·31-s − 63.6·33-s − 117.·35-s − 85.9·37-s + 236.·39-s + 159.·41-s − 92.5·43-s + 110.·45-s + 240.·47-s + 49·49-s + 739.·51-s + 452.·53-s − 185.·55-s + ⋯ |
| L(s) = 1 | + 1.11·3-s + 1.50·5-s − 0.377·7-s + 0.241·9-s − 0.301·11-s + 0.871·13-s + 1.67·15-s + 1.82·17-s + 0.257·19-s − 0.421·21-s + 0.453·23-s + 1.27·25-s − 0.844·27-s − 1.49·29-s + 0.530·31-s − 0.335·33-s − 0.569·35-s − 0.382·37-s + 0.971·39-s + 0.607·41-s − 0.328·43-s + 0.364·45-s + 0.745·47-s + 0.142·49-s + 2.03·51-s + 1.17·53-s − 0.454·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.555195786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.555195786\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 + 11T \) |
| good | 3 | \( 1 - 5.79T + 27T^{2} \) |
| 5 | \( 1 - 16.8T + 125T^{2} \) |
| 13 | \( 1 - 40.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 127.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 21.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 50.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 233.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 91.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 85.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 159.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 92.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 240.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 452.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 49.4T + 2.05e5T^{2} \) |
| 61 | \( 1 - 629.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 575.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 192.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 217.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.13e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 511.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 440.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.10e3T + 9.12e5T^{2} \) |
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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
−9.400232069073921387462452932254, −8.659614204773069165908742314939, −7.85549443576154937820447085162, −6.92389791443490176188365852534, −5.74403612503146087993983517418, −5.48584105335078678294844725411, −3.79632005786992597165934871039, −3.02136106652875018723563173285, −2.14687741760503508135757288265, −1.10797046194882059888836059085,
1.10797046194882059888836059085, 2.14687741760503508135757288265, 3.02136106652875018723563173285, 3.79632005786992597165934871039, 5.48584105335078678294844725411, 5.74403612503146087993983517418, 6.92389791443490176188365852534, 7.85549443576154937820447085162, 8.659614204773069165908742314939, 9.400232069073921387462452932254
