L(s) = 1 | − 5-s + 7-s + 3·11-s + 2·13-s + 2·17-s + 5·19-s + 3·23-s − 4·25-s + 6·29-s + 3·31-s − 35-s − 9·37-s + 3·41-s − 8·43-s + 4·47-s + 49-s + 12·53-s − 3·55-s − 12·59-s + 4·61-s − 2·65-s + 2·67-s − 71-s + 3·77-s + 4·79-s − 6·83-s − 2·85-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.377·7-s + 0.904·11-s + 0.554·13-s + 0.485·17-s + 1.14·19-s + 0.625·23-s − 4/5·25-s + 1.11·29-s + 0.538·31-s − 0.169·35-s − 1.47·37-s + 0.468·41-s − 1.21·43-s + 0.583·47-s + 1/7·49-s + 1.64·53-s − 0.404·55-s − 1.56·59-s + 0.512·61-s − 0.248·65-s + 0.244·67-s − 0.118·71-s + 0.341·77-s + 0.450·79-s − 0.658·83-s − 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.271426870\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.271426870\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−8.165337162817333606821916021574, −7.30691855078910226200071670466, −6.79818045825407078335608178338, −5.90523052358193591897133856067, −5.21405749423837514684479542814, −4.37677487707664520459240091880, −3.64190608377348561234163267928, −2.95314183315036339474647477563, −1.66778541810827258749144620151, −0.851869453680430204125556147557,
0.851869453680430204125556147557, 1.66778541810827258749144620151, 2.95314183315036339474647477563, 3.64190608377348561234163267928, 4.37677487707664520459240091880, 5.21405749423837514684479542814, 5.90523052358193591897133856067, 6.79818045825407078335608178338, 7.30691855078910226200071670466, 8.165337162817333606821916021574