L(s) = 1 | − 2·3-s + 5-s + 2·7-s + 9-s − 2·13-s − 2·15-s − 6·17-s + 4·19-s − 4·21-s − 6·23-s + 25-s + 4·27-s − 6·29-s + 4·31-s + 2·35-s + 2·37-s + 4·39-s − 6·41-s + 10·43-s + 45-s − 3·49-s + 12·51-s − 6·53-s − 8·57-s + 12·59-s + 2·61-s + 2·63-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.554·13-s − 0.516·15-s − 1.45·17-s + 0.917·19-s − 0.872·21-s − 1.25·23-s + 1/5·25-s + 0.769·27-s − 1.11·29-s + 0.718·31-s + 0.338·35-s + 0.328·37-s + 0.640·39-s − 0.937·41-s + 1.52·43-s + 0.149·45-s − 3/7·49-s + 1.68·51-s − 0.824·53-s − 1.05·57-s + 1.56·59-s + 0.256·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9951632622\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9951632622\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 47 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.64403989056791, −14.10766177325342, −13.68703067408208, −13.04144959911860, −12.55315129202397, −11.88537222046621, −11.50592280876449, −11.21486389860933, −10.55488003851877, −10.07407178547446, −9.527677349471193, −8.870839652093621, −8.348876718134038, −7.579395079067937, −7.201658492352857, −6.291889046095977, −6.142679555836660, −5.342887962264324, −4.957632571631969, −4.420563178727819, −3.702816371723760, −2.661606825648993, −2.095651017991132, −1.335934726509509, −0.3938248266010835,
0.3938248266010835, 1.335934726509509, 2.095651017991132, 2.661606825648993, 3.702816371723760, 4.420563178727819, 4.957632571631969, 5.342887962264324, 6.142679555836660, 6.291889046095977, 7.201658492352857, 7.579395079067937, 8.348876718134038, 8.870839652093621, 9.527677349471193, 10.07407178547446, 10.55488003851877, 11.21486389860933, 11.50592280876449, 11.88537222046621, 12.55315129202397, 13.04144959911860, 13.68703067408208, 14.10766177325342, 14.64403989056791