| L(s) = 1 | + (−0.996 + 1.42i)2-s + (−1.44 + 0.949i)3-s + (−0.347 − 0.955i)4-s + (−1.86 + 1.23i)5-s + (0.0912 − 3.00i)6-s + (0.920 − 0.429i)7-s + (−1.64 − 0.442i)8-s + (1.19 − 2.75i)9-s + (0.0978 − 3.88i)10-s + (−0.759 − 0.904i)11-s + (1.41 + 1.05i)12-s + (−3.16 + 2.21i)13-s + (−0.306 + 1.73i)14-s + (1.52 − 3.56i)15-s + (3.83 − 3.21i)16-s + (0.540 − 0.144i)17-s + ⋯ |
| L(s) = 1 | + (−0.704 + 1.00i)2-s + (−0.836 + 0.548i)3-s + (−0.173 − 0.477i)4-s + (−0.833 + 0.552i)5-s + (0.0372 − 1.22i)6-s + (0.347 − 0.162i)7-s + (−0.583 − 0.156i)8-s + (0.398 − 0.917i)9-s + (0.0309 − 1.22i)10-s + (−0.228 − 0.272i)11-s + (0.407 + 0.304i)12-s + (−0.877 + 0.614i)13-s + (−0.0818 + 0.464i)14-s + (0.393 − 0.919i)15-s + (0.957 − 0.803i)16-s + (0.131 − 0.0351i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.706i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.105440 - 0.254824i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.105440 - 0.254824i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.44 - 0.949i)T \) |
| 5 | \( 1 + (1.86 - 1.23i)T \) |
| good | 2 | \( 1 + (0.996 - 1.42i)T + (-0.684 - 1.87i)T^{2} \) |
| 7 | \( 1 + (-0.920 + 0.429i)T + (4.49 - 5.36i)T^{2} \) |
| 11 | \( 1 + (0.759 + 0.904i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (3.16 - 2.21i)T + (4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (-0.540 + 0.144i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (3.83 - 2.21i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.96 - 4.21i)T + (-14.7 - 17.6i)T^{2} \) |
| 29 | \( 1 + (-1.44 - 8.17i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (7.51 - 2.73i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-1.42 - 5.30i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-5.46 - 0.964i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.754 - 0.0660i)T + (42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (5.19 + 11.1i)T + (-30.2 + 36.0i)T^{2} \) |
| 53 | \( 1 + (-5.40 + 5.40i)T - 53iT^{2} \) |
| 59 | \( 1 + (6.20 + 5.20i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-10.5 - 3.83i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.906 - 1.29i)T + (-22.9 + 62.9i)T^{2} \) |
| 71 | \( 1 + (-6.95 - 4.01i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.36 + 5.11i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (6.62 - 1.16i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.245 - 0.172i)T + (28.3 + 77.9i)T^{2} \) |
| 89 | \( 1 + (-5.41 - 9.38i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.446 - 5.10i)T + (-95.5 - 16.8i)T^{2} \) |
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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
−14.53282752081748845275245884440, −12.53699891397034671261016867417, −11.69245754807566124722852048980, −10.73621834978926897193321873933, −9.666454959411447315771892814257, −8.448246576385609425989625477974, −7.34185533295428622458970200223, −6.55828431432780344990695342071, −5.17987019288259364948235777674, −3.65891275288904605984812008215,
0.37804767055153370685503096910, 2.27210184511957436274568513730, 4.49547972163373573673519659409, 5.84353279295984765286966088576, 7.48895132101598761225492749557, 8.391031469246527431996723677022, 9.715972018341550975063610033492, 10.82154802466738611317572342547, 11.46447225205307383774984215436, 12.45094973718583072275415478037
