L(s) = 1 | + 2·7-s + 19-s − 25-s − 2·43-s + 3·49-s − 2·61-s + 2·73-s + ⋯ |
L(s) = 1 | + 2·7-s + 19-s − 25-s − 2·43-s + 3·49-s − 2·61-s + 2·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.551353553\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.551353553\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + T^{2} \) |
| 7 | \( ( 1 - T )^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 + T )^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−8.904324065224969940848831554658, −8.053917425698963362470004563211, −7.76799917031317033328981714321, −6.85091166224273128853735910751, −5.73837352631937928860729675054, −5.07995281953347273668027020258, −4.44007440509091871530622454992, −3.41344110337921118097406721138, −2.13548163632167591855529801716, −1.33098930515104745204770378873,
1.33098930515104745204770378873, 2.13548163632167591855529801716, 3.41344110337921118097406721138, 4.44007440509091871530622454992, 5.07995281953347273668027020258, 5.73837352631937928860729675054, 6.85091166224273128853735910751, 7.76799917031317033328981714321, 8.053917425698963362470004563211, 8.904324065224969940848831554658