L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s − 11-s − 12-s + 16-s + 2·17-s − 22-s − 24-s + 25-s + 27-s + 32-s + 33-s + 2·34-s − 41-s + 2·43-s − 44-s − 48-s + 49-s + 50-s − 2·51-s + 54-s − 59-s + 64-s + 66-s + ⋯ |
L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s − 11-s − 12-s + 16-s + 2·17-s − 22-s − 24-s + 25-s + 27-s + 32-s + 33-s + 2·34-s − 41-s + 2·43-s − 44-s − 48-s + 49-s + 50-s − 2·51-s + 54-s − 59-s + 64-s + 66-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.705799103\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.705799103\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + T + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( ( 1 - T )^{2} \) |
| 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
−8.898963405887509001553324096605, −7.83181899755550511046916947971, −7.36014840530821754841287675413, −6.36988175119366227320900827138, −5.66370064214360302175601493351, −5.27810207358450928973491603061, −4.48462657024718892245510050291, −3.33473769393724325023018227273, −2.63083549380227883843303382695, −1.14489684047718962527384527048,
1.14489684047718962527384527048, 2.63083549380227883843303382695, 3.33473769393724325023018227273, 4.48462657024718892245510050291, 5.27810207358450928973491603061, 5.66370064214360302175601493351, 6.36988175119366227320900827138, 7.36014840530821754841287675413, 7.83181899755550511046916947971, 8.898963405887509001553324096605