L(s) = 1 | + (1.12 − 1.40i)5-s + (−0.900 − 0.433i)9-s + (−0.400 + 0.193i)13-s + 1.24·17-s + (−0.500 − 2.19i)25-s + (0.900 − 0.433i)29-s + (−0.400 − 0.193i)37-s − 0.445·41-s + (−1.62 + 0.781i)45-s + (−0.900 − 0.433i)49-s + (−0.777 + 0.974i)53-s + (0.277 − 1.21i)61-s + (−0.178 + 0.781i)65-s + (0.777 + 0.974i)73-s + (0.623 + 0.781i)81-s + ⋯ |
L(s) = 1 | + (1.12 − 1.40i)5-s + (−0.900 − 0.433i)9-s + (−0.400 + 0.193i)13-s + 1.24·17-s + (−0.500 − 2.19i)25-s + (0.900 − 0.433i)29-s + (−0.400 − 0.193i)37-s − 0.445·41-s + (−1.62 + 0.781i)45-s + (−0.900 − 0.433i)49-s + (−0.777 + 0.974i)53-s + (0.277 − 1.21i)61-s + (−0.178 + 0.781i)65-s + (0.777 + 0.974i)73-s + (0.623 + 0.781i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.288888866\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.288888866\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 + (-0.900 + 0.433i)T \) |
good | 3 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 5 | \( 1 + (-1.12 + 1.40i)T + (-0.222 - 0.974i)T^{2} \) |
| 7 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 11 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 13 | \( 1 + (0.400 - 0.193i)T + (0.623 - 0.781i)T^{2} \) |
| 17 | \( 1 - 1.24T + T^{2} \) |
| 19 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 23 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 31 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 37 | \( 1 + (0.400 + 0.193i)T + (0.623 + 0.781i)T^{2} \) |
| 41 | \( 1 + 0.445T + T^{2} \) |
| 43 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 47 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 53 | \( 1 + (0.777 - 0.974i)T + (-0.222 - 0.974i)T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + (-0.277 + 1.21i)T + (-0.900 - 0.433i)T^{2} \) |
| 67 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 71 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 73 | \( 1 + (-0.777 - 0.974i)T + (-0.222 + 0.974i)T^{2} \) |
| 79 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 83 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 89 | \( 1 + (1.12 - 1.40i)T + (-0.222 - 0.974i)T^{2} \) |
| 97 | \( 1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2} \) |
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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
−9.473003832949264127401230254894, −8.485866928906442214636313954699, −8.053469243132613273779483288832, −6.72420189309195261509417827580, −5.88380350390551735147761391206, −5.32300274845902833745863237052, −4.58032189240322875801189173787, −3.30840804744627297131442995630, −2.16319791149616451417262272536, −1.00529995911701544077670993493,
1.79133123648937081140319502053, 2.83709722240483105810157874761, 3.31217466604819727718733129769, 4.94719996369427809411741744011, 5.69004735769575197629029936385, 6.33710375767659731649440012204, 7.14869728700933275009506880421, 7.932690538788205482651195851311, 8.844676183254137268957967537114, 9.887924595573951873047106980895