L(s) = 1 | + 2-s − 8-s + 9-s − 13-s − 16-s − 17-s + 18-s − 19-s − 23-s + 25-s − 26-s + 31-s − 34-s + 37-s − 38-s + 41-s − 43-s − 46-s + 49-s + 50-s + 59-s − 61-s + 62-s + 64-s + 67-s − 2·71-s − 72-s + ⋯ |
L(s) = 1 | + 2-s − 8-s + 9-s − 13-s − 16-s − 17-s + 18-s − 19-s − 23-s + 25-s − 26-s + 31-s − 34-s + 37-s − 38-s + 41-s − 43-s − 46-s + 49-s + 50-s + 59-s − 61-s + 62-s + 64-s + 67-s − 2·71-s − 72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 247 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 247 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.029618125\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.029618125\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 - T + T^{2} \) |
| 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T + T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( ( 1 + T )^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 + T )^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−12.59029991357469639608547165803, −11.68800533019086439591178658368, −10.42923415355319969643848715721, −9.510547797045711745258007114466, −8.432364021069059372983719053810, −7.07573539214418944329288397203, −6.12485481031552185575926253644, −4.74163982898761448355552412706, −4.15769700191559676645258591419, −2.50768123502979764761201402140,
2.50768123502979764761201402140, 4.15769700191559676645258591419, 4.74163982898761448355552412706, 6.12485481031552185575926253644, 7.07573539214418944329288397203, 8.432364021069059372983719053810, 9.510547797045711745258007114466, 10.42923415355319969643848715721, 11.68800533019086439591178658368, 12.59029991357469639608547165803