| L(s) = 1 | − 2·3-s − 3·7-s + 9-s − 2·17-s + 2·19-s + 6·21-s + 9·23-s + 4·27-s + 2·29-s − 3·31-s − 12·37-s + 7·41-s − 8·43-s − 11·47-s + 2·49-s + 4·51-s − 8·53-s − 4·57-s + 10·59-s + 10·61-s − 3·63-s + 10·67-s − 18·69-s − 8·71-s − 11·73-s + 8·79-s − 11·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.13·7-s + 1/3·9-s − 0.485·17-s + 0.458·19-s + 1.30·21-s + 1.87·23-s + 0.769·27-s + 0.371·29-s − 0.538·31-s − 1.97·37-s + 1.09·41-s − 1.21·43-s − 1.60·47-s + 2/7·49-s + 0.560·51-s − 1.09·53-s − 0.529·57-s + 1.30·59-s + 1.28·61-s − 0.377·63-s + 1.22·67-s − 2.16·69-s − 0.949·71-s − 1.28·73-s + 0.900·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−13.13327923442316, −12.82775563469647, −12.47248452092412, −11.76732383249205, −11.52832512031215, −11.00675934826234, −10.59736417238172, −10.09209481599400, −9.640156078145873, −9.126800148505769, −8.649520735651016, −8.167104240680745, −7.264551230908155, −6.959069983804375, −6.515563669235576, −6.210324136440998, −5.373744313773483, −5.184269178965830, −4.703955079428835, −3.880766592130676, −3.242870284362390, −2.994783950653909, −2.106530422748615, −1.314410724236493, −0.6141462084416108, 0,
0.6141462084416108, 1.314410724236493, 2.106530422748615, 2.994783950653909, 3.242870284362390, 3.880766592130676, 4.703955079428835, 5.184269178965830, 5.373744313773483, 6.210324136440998, 6.515563669235576, 6.959069983804375, 7.264551230908155, 8.167104240680745, 8.649520735651016, 9.126800148505769, 9.640156078145873, 10.09209481599400, 10.59736417238172, 11.00675934826234, 11.52832512031215, 11.76732383249205, 12.47248452092412, 12.82775563469647, 13.13327923442316
