L(s) = 1 | + (−0.841 − 0.540i)2-s + (0.415 + 0.909i)4-s + (0.959 + 0.281i)5-s + (1.10 − 1.27i)7-s + (0.142 − 0.989i)8-s + (−0.959 + 0.281i)9-s + (−0.654 − 0.755i)10-s + (0.698 − 0.449i)11-s + (−1.25 − 1.45i)13-s + (−1.61 + 0.474i)14-s + (−0.654 + 0.755i)16-s + (0.959 + 0.281i)18-s + (−0.118 − 0.258i)19-s + (0.142 + 0.989i)20-s − 0.830·22-s + (0.142 + 0.989i)23-s + ⋯ |
L(s) = 1 | + (−0.841 − 0.540i)2-s + (0.415 + 0.909i)4-s + (0.959 + 0.281i)5-s + (1.10 − 1.27i)7-s + (0.142 − 0.989i)8-s + (−0.959 + 0.281i)9-s + (−0.654 − 0.755i)10-s + (0.698 − 0.449i)11-s + (−1.25 − 1.45i)13-s + (−1.61 + 0.474i)14-s + (−0.654 + 0.755i)16-s + (0.959 + 0.281i)18-s + (−0.118 − 0.258i)19-s + (0.142 + 0.989i)20-s − 0.830·22-s + (0.142 + 0.989i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8236260225\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8236260225\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.841 + 0.540i)T \) |
| 5 | \( 1 + (-0.959 - 0.281i)T \) |
| 23 | \( 1 + (-0.142 - 0.989i)T \) |
good | 3 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 7 | \( 1 + (-1.10 + 1.27i)T + (-0.142 - 0.989i)T^{2} \) |
| 11 | \( 1 + (-0.698 + 0.449i)T + (0.415 - 0.909i)T^{2} \) |
| 13 | \( 1 + (1.25 + 1.45i)T + (-0.142 + 0.989i)T^{2} \) |
| 17 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 19 | \( 1 + (0.118 + 0.258i)T + (-0.654 + 0.755i)T^{2} \) |
| 29 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 31 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 37 | \( 1 + (1.25 - 0.368i)T + (0.841 - 0.540i)T^{2} \) |
| 41 | \( 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2} \) |
| 43 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 47 | \( 1 - 1.30T + T^{2} \) |
| 53 | \( 1 + (1.25 - 1.45i)T + (-0.142 - 0.989i)T^{2} \) |
| 59 | \( 1 + (-0.857 - 0.989i)T + (-0.142 + 0.989i)T^{2} \) |
| 61 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 67 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 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 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 89 | \( 1 + (-0.186 - 1.29i)T + (-0.959 + 0.281i)T^{2} \) |
| 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
−10.32471857266608350169308740832, −9.407036779337117613470977337887, −8.535773169884765604853434373273, −7.67641059792029888920051317497, −7.10849618653490737028570690126, −5.82176183896143526615731593008, −4.84237651594603408284491071537, −3.45100333513634588005812382985, −2.45662482800105307988258129024, −1.16848228591320344769332871586,
1.84678679241168711157305531788, 2.36578285925656073906604677194, 4.70393690296750076035881233063, 5.33365799701837211151246058985, 6.22634920555107720527069463921, 6.95944600657765413668972915109, 8.190107801352534399616678612626, 8.981817048340912033484518406384, 9.191143522853183872551653511089, 10.14221445359935638123308576303