| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s − 11-s − 14-s + 16-s − 20-s − 22-s − 28-s + 2·29-s − 31-s + 32-s + 35-s − 40-s − 44-s − 53-s + 55-s − 56-s + 2·58-s + 2·59-s − 62-s + 64-s + 70-s − 73-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s − 11-s − 14-s + 16-s − 20-s − 22-s − 28-s + 2·29-s − 31-s + 32-s + 35-s − 40-s − 44-s − 53-s + 55-s − 56-s + 2·58-s + 2·59-s − 62-s + 64-s + 70-s − 73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.001867067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.001867067\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + T + T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( ( 1 - T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )^{2} \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−12.61982756847027524258530645604, −11.83752986042266184917870417479, −10.83952224928216910473933489746, −9.926746138506418869471850693025, −8.304816372943545411327964714273, −7.34271926859890419731453562698, −6.34543035073293253125567470780, −5.07819128334573110974618909847, −3.85183234514156199639611073758, −2.79656583583197594606532856647,
2.79656583583197594606532856647, 3.85183234514156199639611073758, 5.07819128334573110974618909847, 6.34543035073293253125567470780, 7.34271926859890419731453562698, 8.304816372943545411327964714273, 9.926746138506418869471850693025, 10.83952224928216910473933489746, 11.83752986042266184917870417479, 12.61982756847027524258530645604
