| L(s) = 1 | − 3.27·2-s + 2.75·4-s + 17.5·5-s + 26.6·7-s + 17.2·8-s − 57.5·10-s − 21.4·11-s − 87.5·14-s − 78.4·16-s − 83.9·17-s + 77.1·19-s + 48.2·20-s + 70.1·22-s − 142.·23-s + 182.·25-s + 73.4·28-s − 134.·29-s − 122.·31-s + 119.·32-s + 275.·34-s + 468.·35-s + 222.·37-s − 252.·38-s + 301.·40-s + 198.·41-s + 154.·43-s − 58.8·44-s + ⋯ |
| L(s) = 1 | − 1.15·2-s + 0.343·4-s + 1.56·5-s + 1.44·7-s + 0.760·8-s − 1.81·10-s − 0.586·11-s − 1.67·14-s − 1.22·16-s − 1.19·17-s + 0.931·19-s + 0.539·20-s + 0.680·22-s − 1.28·23-s + 1.46·25-s + 0.495·28-s − 0.859·29-s − 0.710·31-s + 0.660·32-s + 1.38·34-s + 2.26·35-s + 0.989·37-s − 1.07·38-s + 1.19·40-s + 0.755·41-s + 0.548·43-s − 0.201·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.731123036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.731123036\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 3.27T + 8T^{2} \) |
| 5 | \( 1 - 17.5T + 125T^{2} \) |
| 7 | \( 1 - 26.6T + 343T^{2} \) |
| 11 | \( 1 + 21.4T + 1.33e3T^{2} \) |
| 17 | \( 1 + 83.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 77.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 142.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 134.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 122.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 222.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 198.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 154.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 78.7T + 1.03e5T^{2} \) |
| 53 | \( 1 - 477.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 42.9T + 2.05e5T^{2} \) |
| 61 | \( 1 - 496.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 484.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 382.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 193.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.04e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 861.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 967.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 591.T + 9.12e5T^{2} \) |
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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
−9.356719185524293968437475339887, −8.327968303696890063457261204584, −7.80717406097310269301191776174, −6.90104484342265511047926124291, −5.75107760180436472336145051408, −5.13947947755596359639685768273, −4.19464452831454166136577642030, −2.25560464040033717588463149438, −1.90265759062467063737005529845, −0.78768445496912214744739203245,
0.78768445496912214744739203245, 1.90265759062467063737005529845, 2.25560464040033717588463149438, 4.19464452831454166136577642030, 5.13947947755596359639685768273, 5.75107760180436472336145051408, 6.90104484342265511047926124291, 7.80717406097310269301191776174, 8.327968303696890063457261204584, 9.356719185524293968437475339887
