| L(s) = 1 | + 2-s + 4-s + 8-s − 11-s + 16-s − 17-s − 19-s − 22-s + 25-s + 32-s − 34-s − 38-s − 41-s − 43-s − 44-s + 49-s + 50-s − 59-s + 64-s − 67-s − 68-s − 73-s − 76-s − 82-s + 2·83-s − 86-s − 88-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s + 8-s − 11-s + 16-s − 17-s − 19-s − 22-s + 25-s + 32-s − 34-s − 38-s − 41-s − 43-s − 44-s + 49-s + 50-s − 59-s + 64-s − 67-s − 68-s − 73-s − 76-s − 82-s + 2·83-s − 86-s − 88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.598301541\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.598301541\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 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 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )^{2} \) |
| 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
−10.74638251768534715971124010278, −10.35692740638567354651264058256, −8.932976995690384528300959834726, −8.005467885314743797522085321079, −7.00209538156467330246099082489, −6.22946753108448932064253691050, −5.13981126561001631297294145990, −4.39577787880565181212977339090, −3.13109369433609716423062418558, −2.07086974203217045980922116658,
2.07086974203217045980922116658, 3.13109369433609716423062418558, 4.39577787880565181212977339090, 5.13981126561001631297294145990, 6.22946753108448932064253691050, 7.00209538156467330246099082489, 8.005467885314743797522085321079, 8.932976995690384528300959834726, 10.35692740638567354651264058256, 10.74638251768534715971124010278
