| L(s) = 1 | − 1.95·2-s − 4.15·4-s − 11.8·5-s + 23.8·8-s + 23.3·10-s − 36.4·11-s + 0.964·13-s − 13.4·16-s − 98.2·17-s − 106.·19-s + 49.4·20-s + 71.3·22-s − 54.3·23-s + 16.3·25-s − 1.89·26-s − 229.·29-s + 127.·31-s − 164.·32-s + 192.·34-s + 311.·37-s + 207.·38-s − 283.·40-s + 419.·41-s + 523.·43-s + 151.·44-s + 106.·46-s − 270.·47-s + ⋯ |
| L(s) = 1 | − 0.692·2-s − 0.519·4-s − 1.06·5-s + 1.05·8-s + 0.736·10-s − 0.998·11-s + 0.0205·13-s − 0.209·16-s − 1.40·17-s − 1.27·19-s + 0.552·20-s + 0.691·22-s − 0.492·23-s + 0.130·25-s − 0.0142·26-s − 1.47·29-s + 0.740·31-s − 0.907·32-s + 0.971·34-s + 1.38·37-s + 0.886·38-s − 1.11·40-s + 1.59·41-s + 1.85·43-s + 0.518·44-s + 0.341·46-s − 0.838·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3791382839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3791382839\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 1.95T + 8T^{2} \) |
| 5 | \( 1 + 11.8T + 125T^{2} \) |
| 11 | \( 1 + 36.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 0.964T + 2.19e3T^{2} \) |
| 17 | \( 1 + 98.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 106.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 54.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 229.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 127.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 311.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 419.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 523.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 270.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 251.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 408.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 860.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 506.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 523.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 629.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 319.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.30e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 348.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 161.T + 9.12e5T^{2} \) |
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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
−10.85304811263181088657386766707, −9.672179120583069373996119016417, −8.824018619929645179858890698931, −7.980038302075092099215989086971, −7.47145749697816320201343253715, −6.06339242529224459019967544407, −4.60767881777008710731432786039, −4.02082141889044503297319971960, −2.30209701323291969294128116791, −0.41954319055443436807373191201,
0.41954319055443436807373191201, 2.30209701323291969294128116791, 4.02082141889044503297319971960, 4.60767881777008710731432786039, 6.06339242529224459019967544407, 7.47145749697816320201343253715, 7.980038302075092099215989086971, 8.824018619929645179858890698931, 9.672179120583069373996119016417, 10.85304811263181088657386766707
