L(s) = 1 | + (−0.153 + 1.07i)3-s + (0.959 − 0.281i)5-s + (0.989 + 1.14i)7-s + (−0.162 − 0.0476i)9-s + (0.153 + 1.07i)15-s + (−1.37 + 0.883i)21-s + (−0.540 − 0.841i)23-s + (0.841 − 0.540i)25-s + (−0.373 + 0.817i)27-s + (−0.797 − 1.74i)29-s + (1.27 + 0.817i)35-s + (−1.25 + 0.368i)41-s + (0.258 − 1.80i)43-s − 0.169·45-s + 0.563·47-s + ⋯ |
L(s) = 1 | + (−0.153 + 1.07i)3-s + (0.959 − 0.281i)5-s + (0.989 + 1.14i)7-s + (−0.162 − 0.0476i)9-s + (0.153 + 1.07i)15-s + (−1.37 + 0.883i)21-s + (−0.540 − 0.841i)23-s + (0.841 − 0.540i)25-s + (−0.373 + 0.817i)27-s + (−0.797 − 1.74i)29-s + (1.27 + 0.817i)35-s + (−1.25 + 0.368i)41-s + (0.258 − 1.80i)43-s − 0.169·45-s + 0.563·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.445733085\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.445733085\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.959 + 0.281i)T \) |
| 23 | \( 1 + (0.540 + 0.841i)T \) |
good | 3 | \( 1 + (0.153 - 1.07i)T + (-0.959 - 0.281i)T^{2} \) |
| 7 | \( 1 + (-0.989 - 1.14i)T + (-0.142 + 0.989i)T^{2} \) |
| 11 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 13 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 17 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 19 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 29 | \( 1 + (0.797 + 1.74i)T + (-0.654 + 0.755i)T^{2} \) |
| 31 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 37 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 41 | \( 1 + (1.25 - 0.368i)T + (0.841 - 0.540i)T^{2} \) |
| 43 | \( 1 + (-0.258 + 1.80i)T + (-0.959 - 0.281i)T^{2} \) |
| 47 | \( 1 - 0.563T + T^{2} \) |
| 53 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 59 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 61 | \( 1 + (-0.118 - 0.822i)T + (-0.959 + 0.281i)T^{2} \) |
| 67 | \( 1 + (1.66 - 1.07i)T + (0.415 - 0.909i)T^{2} \) |
| 71 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 73 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 79 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 83 | \( 1 + (1.89 + 0.557i)T + (0.841 + 0.540i)T^{2} \) |
| 89 | \( 1 + (0.239 - 1.66i)T + (-0.959 - 0.281i)T^{2} \) |
| 97 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
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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
−9.690052450764438965864545658169, −8.821495444742304678787638383527, −8.423056818248937688041328769795, −7.25625252915749256246376191318, −5.98880476706943122174815423515, −5.53556986014992098027614595452, −4.76660639554355887987378909510, −4.04415993290332208602055297269, −2.58138768024164921070491918371, −1.76968567652109816844179979222,
1.34807238197414105560840664360, 1.81989239485359275316367295905, 3.25084152502315664631982213082, 4.43806473789878155493725241975, 5.37171931768850643364114358788, 6.22483660108119053215625012217, 7.08417946948941614662514138237, 7.45865485235614553345340265292, 8.302260250496499263867260341569, 9.324805372262467202573093883310