| L(s) = 1 | + (−0.923 + 0.382i)2-s + (−0.596 − 0.118i)3-s + (0.707 − 0.707i)4-s + (−1.74 − 2.61i)5-s + (0.596 − 0.118i)6-s + (1.59 + 2.10i)7-s + (−0.382 + 0.923i)8-s + (−2.43 − 1.00i)9-s + (2.61 + 1.74i)10-s + (−3.66 + 0.728i)11-s + (−0.505 + 0.337i)12-s + (−2.81 + 2.81i)13-s + (−2.28 − 1.33i)14-s + (0.732 + 1.76i)15-s − i·16-s + (−2.54 − 3.24i)17-s + ⋯ |
| L(s) = 1 | + (−0.653 + 0.270i)2-s + (−0.344 − 0.0684i)3-s + (0.353 − 0.353i)4-s + (−0.782 − 1.17i)5-s + (0.243 − 0.0484i)6-s + (0.603 + 0.797i)7-s + (−0.135 + 0.326i)8-s + (−0.810 − 0.335i)9-s + (0.828 + 0.553i)10-s + (−1.10 + 0.219i)11-s + (−0.145 + 0.0974i)12-s + (−0.779 + 0.779i)13-s + (−0.610 − 0.357i)14-s + (0.189 + 0.456i)15-s − 0.250i·16-s + (−0.617 − 0.786i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0331529 - 0.168411i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0331529 - 0.168411i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.923 - 0.382i)T \) |
| 7 | \( 1 + (-1.59 - 2.10i)T \) |
| 17 | \( 1 + (2.54 + 3.24i)T \) |
| good | 3 | \( 1 + (0.596 + 0.118i)T + (2.77 + 1.14i)T^{2} \) |
| 5 | \( 1 + (1.74 + 2.61i)T + (-1.91 + 4.61i)T^{2} \) |
| 11 | \( 1 + (3.66 - 0.728i)T + (10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (2.81 - 2.81i)T - 13iT^{2} \) |
| 19 | \( 1 + (2.08 + 5.04i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-0.0421 - 0.211i)T + (-21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (1.71 + 2.57i)T + (-11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 + (0.521 - 2.62i)T + (-28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (-0.603 + 3.03i)T + (-34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (4.30 - 6.43i)T + (-15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (3.97 + 1.64i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (-3.26 + 3.26i)T - 47iT^{2} \) |
| 53 | \( 1 + (-3.59 + 1.48i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-7.33 - 3.03i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-1.63 - 1.09i)T + (23.3 + 56.3i)T^{2} \) |
| 67 | \( 1 + 15.5iT - 67T^{2} \) |
| 71 | \( 1 + (-1.58 + 7.96i)T + (-65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-6.68 - 10.0i)T + (-27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-0.195 + 0.0389i)T + (72.9 - 30.2i)T^{2} \) |
| 83 | \( 1 + (-13.2 + 5.48i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (7.68 + 7.68i)T + 89iT^{2} \) |
| 97 | \( 1 + (0.729 - 0.487i)T + (37.1 - 89.6i)T^{2} \) |
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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
−11.71697477698563462374504955061, −10.98410639261762787826399124278, −9.421621438795655366828985509310, −8.741511476841401649783535193253, −8.003226447567208972392287196537, −6.84116573305504194419426590432, −5.32311438043585439020328237425, −4.72969090527163191127673240988, −2.40521361898560855441643721720, −0.16491422206120525742381850638,
2.52017177242478800962028983222, 3.80527591446068860166857100680, 5.41070872264901909165748703469, 6.83180515946788443885435298491, 7.85773908055705982062124879118, 8.278925562769224732974401927008, 10.19878078133519582079601532044, 10.67517077766367822010778322499, 11.21321752731777200335539223902, 12.23575002437242009391261526544
