L(s) = 1 | − 2·3-s − 2·5-s − 7-s + 9-s + 2·11-s − 2·13-s + 4·15-s + 4·19-s + 2·21-s + 4·23-s − 25-s + 4·27-s − 2·29-s + 8·31-s − 4·33-s + 2·35-s + 2·37-s + 4·39-s + 4·43-s − 2·45-s − 12·47-s + 49-s − 14·53-s − 4·55-s − 8·57-s + 12·59-s + 10·61-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s + 1.03·15-s + 0.917·19-s + 0.436·21-s + 0.834·23-s − 1/5·25-s + 0.769·27-s − 0.371·29-s + 1.43·31-s − 0.696·33-s + 0.338·35-s + 0.328·37-s + 0.640·39-s + 0.609·43-s − 0.298·45-s − 1.75·47-s + 1/7·49-s − 1.92·53-s − 0.539·55-s − 1.05·57-s + 1.56·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8107784347\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8107784347\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−15.94598825674874, −15.66540840594199, −14.87852515604695, −14.37905913483692, −13.75807937839698, −12.88724651370797, −12.55843172617478, −11.81630121752600, −11.52253801843719, −11.19503957099078, −10.39283817582267, −9.699893848600391, −9.343218833800638, −8.313581118506352, −7.937281611542202, −7.043094233052944, −6.679047673019052, −6.041089734273581, −5.222962080048572, −4.830469806307065, −4.003513856706236, −3.336215833968429, −2.540853776649938, −1.227217092623877, −0.4688997943609977,
0.4688997943609977, 1.227217092623877, 2.540853776649938, 3.336215833968429, 4.003513856706236, 4.830469806307065, 5.222962080048572, 6.041089734273581, 6.679047673019052, 7.043094233052944, 7.937281611542202, 8.313581118506352, 9.343218833800638, 9.699893848600391, 10.39283817582267, 11.19503957099078, 11.52253801843719, 11.81630121752600, 12.55843172617478, 12.88724651370797, 13.75807937839698, 14.37905913483692, 14.87852515604695, 15.66540840594199, 15.94598825674874