| L(s) = 1 | + 2i·2-s − 4·4-s − 4.69i·7-s − 8i·8-s − 14.3·11-s − 68.4i·13-s + 9.39·14-s + 16·16-s − 55.4i·17-s − 57.4·19-s − 28.7i·22-s + 64.4i·23-s + 136.·26-s + 18.7i·28-s − 133.·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.253i·7-s − 0.353i·8-s − 0.394·11-s − 1.46i·13-s + 0.179·14-s + 0.250·16-s − 0.791i·17-s − 0.694·19-s − 0.278i·22-s + 0.584i·23-s + 1.03·26-s + 0.126i·28-s − 0.854·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6560701474\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6560701474\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4.69iT - 343T^{2} \) |
| 11 | \( 1 + 14.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 68.4iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 55.4iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 57.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 64.4iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 133.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 24.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 160. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 121.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 57.1iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 433. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 310. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 40.7T + 2.05e5T^{2} \) |
| 61 | \( 1 + 8.54T + 2.26e5T^{2} \) |
| 67 | \( 1 - 474. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 334.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 518. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 951.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 12.1iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 462.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 449. iT - 9.12e5T^{2} \) |
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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
−9.474882927502011325911937042318, −8.664180095077682822187018666222, −7.73660643780923883070542759538, −7.36986104083434822695116579476, −6.21800462939198672739101934079, −5.50666778374703176677527453026, −4.70329594376212024293456169243, −3.61144629274341776294447992425, −2.59292859529125591227034013080, −0.971514529096991959099873975677,
0.17197823897964181006230514923, 1.71287626507528439582135219816, 2.40688455609904696999173450662, 3.73104716305595594235633804086, 4.39269796757633120220152754857, 5.44704321792310517955425979048, 6.37158141276922696551922403822, 7.27999080780648683915326192489, 8.372011027116105444607266666445, 8.938066209691007598748632597828
