| L(s) = 1 | + (0.353 − 2.65i)3-s + (−1.27 + 0.562i)5-s + (−1.92 − 4.13i)7-s + (−4.02 − 1.09i)9-s + (−1.02 − 1.30i)11-s + (1.42 + 2.77i)13-s + (1.04 + 3.57i)15-s + (−2.44 + 0.185i)17-s + (2.03 + 3.06i)19-s + (−11.6 + 3.64i)21-s + (0.849 − 0.337i)23-s + (−2.06 + 2.26i)25-s + (−1.21 + 2.88i)27-s + (−4.52 − 1.79i)29-s + (−2.76 − 0.638i)31-s + ⋯ |
| L(s) = 1 | + (0.204 − 1.53i)3-s + (−0.568 + 0.251i)5-s + (−0.727 − 1.56i)7-s + (−1.34 − 0.363i)9-s + (−0.307 − 0.394i)11-s + (0.394 + 0.769i)13-s + (0.269 + 0.923i)15-s + (−0.591 + 0.0448i)17-s + (0.467 + 0.703i)19-s + (−2.54 + 0.795i)21-s + (0.177 − 0.0704i)23-s + (−0.412 + 0.453i)25-s + (−0.232 + 0.554i)27-s + (−0.840 − 0.334i)29-s + (−0.496 − 0.114i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.128923 + 0.764422i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.128923 + 0.764422i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 + (-3.39 - 12.4i)T \) |
| good | 3 | \( 1 + (-0.353 + 2.65i)T + (-2.89 - 0.785i)T^{2} \) |
| 5 | \( 1 + (1.27 - 0.562i)T + (3.36 - 3.69i)T^{2} \) |
| 7 | \( 1 + (1.92 + 4.13i)T + (-4.51 + 5.35i)T^{2} \) |
| 11 | \( 1 + (1.02 + 1.30i)T + (-2.67 + 10.6i)T^{2} \) |
| 13 | \( 1 + (-1.42 - 2.77i)T + (-7.60 + 10.5i)T^{2} \) |
| 17 | \( 1 + (2.44 - 0.185i)T + (16.8 - 2.56i)T^{2} \) |
| 19 | \( 1 + (-2.03 - 3.06i)T + (-7.35 + 17.5i)T^{2} \) |
| 23 | \( 1 + (-0.849 + 0.337i)T + (16.7 - 15.7i)T^{2} \) |
| 29 | \( 1 + (4.52 + 1.79i)T + (21.0 + 19.9i)T^{2} \) |
| 31 | \( 1 + (2.76 + 0.638i)T + (27.8 + 13.6i)T^{2} \) |
| 37 | \( 1 + (-8.57 + 2.32i)T + (31.9 - 18.7i)T^{2} \) |
| 41 | \( 1 + (5.33 + 2.60i)T + (25.2 + 32.3i)T^{2} \) |
| 43 | \( 1 + (0.373 + 3.93i)T + (-42.2 + 8.08i)T^{2} \) |
| 47 | \( 1 + (-0.147 + 7.77i)T + (-46.9 - 1.77i)T^{2} \) |
| 53 | \( 1 + (-0.188 + 1.09i)T + (-49.9 - 17.7i)T^{2} \) |
| 59 | \( 1 + (10.3 + 0.787i)T + (58.3 + 8.89i)T^{2} \) |
| 61 | \( 1 + (-5.52 - 5.62i)T + (-1.15 + 60.9i)T^{2} \) |
| 67 | \( 1 + (8.24 + 3.64i)T + (45.0 + 49.5i)T^{2} \) |
| 71 | \( 1 + (-11.0 - 1.26i)T + (69.1 + 15.9i)T^{2} \) |
| 73 | \( 1 + (3.43 + 2.57i)T + (20.4 + 70.0i)T^{2} \) |
| 79 | \( 1 + (0.402 + 1.07i)T + (-59.4 + 52.0i)T^{2} \) |
| 83 | \( 1 + (4.94 + 6.85i)T + (-26.2 + 78.7i)T^{2} \) |
| 89 | \( 1 + (-0.517 + 1.77i)T + (-75.0 - 47.8i)T^{2} \) |
| 97 | \( 1 + (-5.32 + 1.23i)T + (87.1 - 42.5i)T^{2} \) |
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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
−10.12201587184457130743627867399, −9.009596339757509572322760584677, −7.905615955379542964210641575746, −7.35725754699816963428087211051, −6.77795245524067419309008709015, −5.89462007111709987957596139503, −4.12464454773029808759940863601, −3.30939200346708601388437590918, −1.77577647522547028000445723807, −0.39245231931398659170018600367,
2.63420515633825528568827593987, 3.44018996270000832392683032614, 4.59039099204919690791787378378, 5.34982432247604290189117662256, 6.27715058462937473062998382819, 7.78759832362716346368103355643, 8.713374684610447801858853583730, 9.318436706780190325746033832241, 9.903856050891183409064801403628, 10.96394489646287876905284030309
