L(s) = 1 | + (−0.900 − 0.433i)2-s + (0.623 + 0.781i)4-s + (0.400 + 0.193i)5-s + (−0.222 − 0.974i)8-s + (−0.222 − 0.974i)9-s + (−0.277 − 0.347i)10-s + (−0.277 + 1.21i)13-s + (−0.222 + 0.974i)16-s − 1.80·17-s + (−0.222 + 0.974i)18-s + (0.0990 + 0.433i)20-s + (−0.499 − 0.626i)25-s + (0.777 − 0.974i)26-s + (−0.222 + 0.974i)29-s + (0.623 − 0.781i)32-s + ⋯ |
L(s) = 1 | + (−0.900 − 0.433i)2-s + (0.623 + 0.781i)4-s + (0.400 + 0.193i)5-s + (−0.222 − 0.974i)8-s + (−0.222 − 0.974i)9-s + (−0.277 − 0.347i)10-s + (−0.277 + 1.21i)13-s + (−0.222 + 0.974i)16-s − 1.80·17-s + (−0.222 + 0.974i)18-s + (0.0990 + 0.433i)20-s + (−0.499 − 0.626i)25-s + (0.777 − 0.974i)26-s + (−0.222 + 0.974i)29-s + (0.623 − 0.781i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4257392580\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4257392580\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.900 + 0.433i)T \) |
| 29 | \( 1 + (0.222 - 0.974i)T \) |
good | 3 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 5 | \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \) |
| 7 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 11 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 13 | \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \) |
| 17 | \( 1 + 1.80T + T^{2} \) |
| 19 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 23 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 31 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 37 | \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \) |
| 41 | \( 1 - 1.24T + T^{2} \) |
| 43 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 47 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 53 | \( 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + (1.12 - 1.40i)T + (-0.222 - 0.974i)T^{2} \) |
| 67 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 71 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (-1.62 + 0.781i)T + (0.623 - 0.781i)T^{2} \) |
| 79 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 83 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 89 | \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \) |
| 97 | \( 1 + (-0.777 - 0.974i)T + (-0.222 + 0.974i)T^{2} \) |
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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
−13.63790628989257083554662323178, −12.43466690849536647688375751949, −11.54254152352514357558457178863, −10.59330318001536347634774278641, −9.319946352849133700748968440668, −8.837159895492262790052303301048, −7.16357847299187705940182612447, −6.29445753897241092991800488921, −4.05244636358745114329875164559, −2.26607837765745910984800191378,
2.31827131435495899951581404568, 4.99363779169325571996143279210, 6.13947680312980219114799289374, 7.52818446314234848157134324994, 8.449234357076432716096816256328, 9.568815941000766169741361940621, 10.59506596567294664753546040716, 11.40820875245916494763818012549, 13.02039422204479340809031940920, 13.87815191103923083814203492603