| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 12-s + 13-s + 14-s + 16-s − 17-s − 19-s − 21-s − 23-s − 24-s + 25-s − 26-s − 27-s − 28-s + 29-s − 32-s + 34-s − 2·37-s + 38-s + 39-s + 42-s + 46-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 12-s + 13-s + 14-s + 16-s − 17-s − 19-s − 21-s − 23-s − 24-s + 25-s − 26-s − 27-s − 28-s + 29-s − 32-s + 34-s − 2·37-s + 38-s + 39-s + 42-s + 46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5255734463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5255734463\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - T + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 + T )^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−13.27399541754351592744446478828, −12.20286584791493355118555394928, −10.89692957890453178097024652546, −10.03706376682674974881302024345, −8.724373722575583812286798578832, −8.616744569991860759400358457809, −7.04273464292366857250117884391, −6.10266027944254780694352976077, −3.65604067108832041139853999554, −2.38472062956344778846731955046,
2.38472062956344778846731955046, 3.65604067108832041139853999554, 6.10266027944254780694352976077, 7.04273464292366857250117884391, 8.616744569991860759400358457809, 8.724373722575583812286798578832, 10.03706376682674974881302024345, 10.89692957890453178097024652546, 12.20286584791493355118555394928, 13.27399541754351592744446478828
