| L(s) = 1 | + 4-s − 2·7-s + 16-s − 19-s − 25-s − 2·28-s + 2·43-s + 3·49-s − 2·61-s + 64-s + 2·73-s − 76-s − 100-s − 2·112-s + ⋯ |
| L(s) = 1 | + 4-s − 2·7-s + 16-s − 19-s − 25-s − 2·28-s + 2·43-s + 3·49-s − 2·61-s + 64-s + 2·73-s − 76-s − 100-s − 2·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6894863364\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6894863364\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 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
−12.77174751657485658890715096643, −12.21494054797840610775226859472, −10.92993757108957066673568740756, −10.07541152140568924855662867611, −9.143074012490020203281745472011, −7.61832565927716848254810692489, −6.55795206210644699740063266710, −5.92525912730615074752331059629, −3.77449706534207662124002521236, −2.55662379043524608621887501562,
2.55662379043524608621887501562, 3.77449706534207662124002521236, 5.92525912730615074752331059629, 6.55795206210644699740063266710, 7.61832565927716848254810692489, 9.143074012490020203281745472011, 10.07541152140568924855662867611, 10.92993757108957066673568740756, 12.21494054797840610775226859472, 12.77174751657485658890715096643
