| L(s) = 1 | + 2.78·2-s − 0.244·4-s + 20.8·5-s − 22.9·8-s + 58.0·10-s + 65.7·11-s − 50.9·13-s − 61.9·16-s − 46.1·17-s + 62.1·19-s − 5.09·20-s + 183.·22-s + 125.·23-s + 308.·25-s − 141.·26-s − 168.·29-s + 187.·31-s + 11.0·32-s − 128.·34-s − 70.0·37-s + 173.·38-s − 478.·40-s + 188.·41-s + 239.·43-s − 16.0·44-s + 349.·46-s + 591.·47-s + ⋯ |
| L(s) = 1 | + 0.984·2-s − 0.0305·4-s + 1.86·5-s − 1.01·8-s + 1.83·10-s + 1.80·11-s − 1.08·13-s − 0.968·16-s − 0.658·17-s + 0.750·19-s − 0.0569·20-s + 1.77·22-s + 1.13·23-s + 2.47·25-s − 1.06·26-s − 1.08·29-s + 1.08·31-s + 0.0611·32-s − 0.647·34-s − 0.311·37-s + 0.738·38-s − 1.89·40-s + 0.717·41-s + 0.849·43-s − 0.0551·44-s + 1.11·46-s + 1.83·47-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
| \(L(2)\) |
\(\approx\) |
\(5.015988501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.015988501\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 2.78T + 8T^{2} \) |
| 5 | \( 1 - 20.8T + 125T^{2} \) |
| 11 | \( 1 - 65.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 50.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 46.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 62.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 125.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 168.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 187.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 70.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 188.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 239.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 591.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 464.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 325.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 261.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 340.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 752.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 245.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 546.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 144.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 603.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.35e3T + 9.12e5T^{2} \) |
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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
−9.161832431642491045494564647593, −9.039285798227110830434576582590, −7.25169433135849502073734553840, −6.44337805383435953594737042047, −5.87415210182200023725175113365, −5.04736806075532183373433863722, −4.32014485989271344474754956229, −3.10119958152249523559508703292, −2.19222594412830502245172157774, −1.03140266686704113044661630908,
1.03140266686704113044661630908, 2.19222594412830502245172157774, 3.10119958152249523559508703292, 4.32014485989271344474754956229, 5.04736806075532183373433863722, 5.87415210182200023725175113365, 6.44337805383435953594737042047, 7.25169433135849502073734553840, 9.039285798227110830434576582590, 9.161832431642491045494564647593
