| L(s) = 1 | − 5.15·2-s + 18.5·4-s + 17.0·5-s − 54.6·8-s − 87.7·10-s + 30.3·11-s + 89.5·13-s + 132.·16-s + 13.4·17-s + 5.37·19-s + 316.·20-s − 156.·22-s − 167.·23-s + 164.·25-s − 461.·26-s + 135.·29-s + 18.9·31-s − 248.·32-s − 69.1·34-s + 402.·37-s − 27.7·38-s − 929.·40-s − 434.·41-s + 53.1·43-s + 564.·44-s + 863.·46-s − 155.·47-s + ⋯ |
| L(s) = 1 | − 1.82·2-s + 2.32·4-s + 1.52·5-s − 2.41·8-s − 2.77·10-s + 0.831·11-s + 1.91·13-s + 2.07·16-s + 0.191·17-s + 0.0648·19-s + 3.53·20-s − 1.51·22-s − 1.51·23-s + 1.31·25-s − 3.48·26-s + 0.864·29-s + 0.109·31-s − 1.37·32-s − 0.348·34-s + 1.78·37-s − 0.118·38-s − 3.67·40-s − 1.65·41-s + 0.188·43-s + 1.93·44-s + 2.76·46-s − 0.481·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.583907067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.583907067\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 5.15T + 8T^{2} \) |
| 5 | \( 1 - 17.0T + 125T^{2} \) |
| 11 | \( 1 - 30.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 89.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 13.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 5.37T + 6.85e3T^{2} \) |
| 23 | \( 1 + 167.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 135.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 18.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 402.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 434.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 53.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + 155.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 301.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 412.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 571.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 820.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 8.95T + 3.57e5T^{2} \) |
| 73 | \( 1 + 21.9T + 3.89e5T^{2} \) |
| 79 | \( 1 - 619.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 259.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 484.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 252.T + 9.12e5T^{2} \) |
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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
−9.326880276808664659892034934584, −8.545220716226321437845464531334, −8.073286270866291712210857159836, −6.66235505984436135115370742789, −6.36988191177389327727810923945, −5.58626466266196626954064722800, −3.81535765385454747994025872823, −2.47162418026906036769262843872, −1.58168507727422916655856884221, −0.917592372280660955496232910496,
0.917592372280660955496232910496, 1.58168507727422916655856884221, 2.47162418026906036769262843872, 3.81535765385454747994025872823, 5.58626466266196626954064722800, 6.36988191177389327727810923945, 6.66235505984436135115370742789, 8.073286270866291712210857159836, 8.545220716226321437845464531334, 9.326880276808664659892034934584
