| L(s) = 1 | + 3-s − 5-s + 9-s − 11-s − 13-s − 15-s + 17-s − 19-s − 23-s + 27-s + 2·29-s − 33-s − 39-s − 41-s − 43-s − 45-s + 49-s + 51-s + 55-s − 57-s + 65-s + 2·67-s − 69-s + 2·71-s + 81-s − 85-s + 2·87-s + ⋯ |
| L(s) = 1 | + 3-s − 5-s + 9-s − 11-s − 13-s − 15-s + 17-s − 19-s − 23-s + 27-s + 2·29-s − 33-s − 39-s − 41-s − 43-s − 45-s + 49-s + 51-s + 55-s − 57-s + 65-s + 2·67-s − 69-s + 2·71-s + 81-s − 85-s + 2·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7863288172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7863288172\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + T + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( ( 1 - T )^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )^{2} \) |
| 71 | \( ( 1 - T )^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 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
−12.55293235020929802988028909026, −11.97716062598023379286844566197, −10.47631458520706009380759975999, −9.805222121350042209772523186414, −8.284936952968746021179779787637, −7.965202984910309898471426703055, −6.82424899172616989471966838318, −4.97602311316391520571027185228, −3.79943974802985091918625662732, −2.51256673938332341097950717807,
2.51256673938332341097950717807, 3.79943974802985091918625662732, 4.97602311316391520571027185228, 6.82424899172616989471966838318, 7.965202984910309898471426703055, 8.284936952968746021179779787637, 9.805222121350042209772523186414, 10.47631458520706009380759975999, 11.97716062598023379286844566197, 12.55293235020929802988028909026
