| L(s) = 1 | + (0.841 + 0.540i)2-s + (0.142 + 0.989i)3-s + (0.415 + 0.909i)4-s + (0.186 + 0.0547i)5-s + (−0.415 + 0.909i)6-s + (−1.27 + 1.47i)7-s + (−0.142 + 0.989i)8-s + (−0.959 + 0.281i)9-s + (0.127 + 0.146i)10-s + (5.30 − 3.40i)11-s + (−0.841 + 0.540i)12-s + (−3.15 − 3.63i)13-s + (−1.87 + 0.549i)14-s + (−0.0276 + 0.192i)15-s + (−0.654 + 0.755i)16-s + (−1.33 + 2.91i)17-s + ⋯ |
| L(s) = 1 | + (0.594 + 0.382i)2-s + (0.0821 + 0.571i)3-s + (0.207 + 0.454i)4-s + (0.0833 + 0.0244i)5-s + (−0.169 + 0.371i)6-s + (−0.483 + 0.557i)7-s + (−0.0503 + 0.349i)8-s + (−0.319 + 0.0939i)9-s + (0.0402 + 0.0464i)10-s + (1.59 − 1.02i)11-s + (−0.242 + 0.156i)12-s + (−0.874 − 1.00i)13-s + (−0.500 + 0.146i)14-s + (−0.00713 + 0.0496i)15-s + (−0.163 + 0.188i)16-s + (−0.323 + 0.707i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.25420 + 0.782320i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25420 + 0.782320i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.841 - 0.540i)T \) |
| 3 | \( 1 + (-0.142 - 0.989i)T \) |
| 23 | \( 1 + (-3.37 + 3.40i)T \) |
| good | 5 | \( 1 + (-0.186 - 0.0547i)T + (4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (1.27 - 1.47i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-5.30 + 3.40i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (3.15 + 3.63i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.33 - 2.91i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.357 - 0.781i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-2.25 + 4.92i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.144 + 1.00i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (6.61 - 1.94i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-0.932 - 0.273i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.731 - 5.09i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 7.41T + 47T^{2} \) |
| 53 | \( 1 + (1.31 - 1.51i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-5.63 - 6.50i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (1.72 - 11.9i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-5.96 - 3.83i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-0.933 - 0.599i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (5.47 + 11.9i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (0.739 + 0.853i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (3.84 - 1.12i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-1.35 - 9.45i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-1.07 - 0.315i)T + (81.6 + 52.4i)T^{2} \) |
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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
−13.49196172445335313288683922984, −12.38084642126238498756774877648, −11.54995609827969786112823635508, −10.27278888945643772939156205851, −9.131422001742455915267522262254, −8.171439127469892599497152676201, −6.52917999790651131901616438254, −5.69747638130967357176740035943, −4.21918002703971402997536291996, −2.95948536190267993125782890777,
1.80815365556608671869491429685, 3.65212002451216462848031331302, 4.95529860332498646354984784881, 6.77118330627980250363013700903, 7.08965614108222283461710926579, 9.147574332393560428585945449209, 9.822761980145121457631198041670, 11.37867704583409021944116254080, 12.05264907227845944892218626250, 12.96037179337720993608169422866
