| L(s) = 1 | + 3-s − 2·9-s − 3·11-s + 3·13-s + 3·17-s + 2·19-s − 23-s − 5·27-s + 3·29-s + 4·31-s − 3·33-s − 8·37-s + 3·39-s + 10·41-s − 4·43-s + 13·47-s + 3·51-s + 2·57-s + 8·59-s + 10·61-s + 6·67-s − 69-s + 6·73-s + 7·79-s + 81-s + 3·87-s + 10·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s − 0.904·11-s + 0.832·13-s + 0.727·17-s + 0.458·19-s − 0.208·23-s − 0.962·27-s + 0.557·29-s + 0.718·31-s − 0.522·33-s − 1.31·37-s + 0.480·39-s + 1.56·41-s − 0.609·43-s + 1.89·47-s + 0.420·51-s + 0.264·57-s + 1.04·59-s + 1.28·61-s + 0.733·67-s − 0.120·69-s + 0.702·73-s + 0.787·79-s + 1/9·81-s + 0.321·87-s + 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.475979417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.475979417\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−13.02933815842981, −12.43056602787048, −12.12814449684217, −11.48943927193593, −11.12164700595269, −10.59278525274928, −10.11085965322266, −9.749907717097335, −9.046438013783274, −8.696176439422418, −8.249931191305922, −7.843612614369808, −7.389443107728796, −6.793409255841564, −6.146623597138799, −5.699580247534923, −5.271311152353508, −4.769114542100877, −3.842063758551388, −3.659989327703675, −2.949429330587402, −2.493738714717895, −1.999588779896385, −1.037236652263001, −0.5632387454151324,
0.5632387454151324, 1.037236652263001, 1.999588779896385, 2.493738714717895, 2.949429330587402, 3.659989327703675, 3.842063758551388, 4.769114542100877, 5.271311152353508, 5.699580247534923, 6.146623597138799, 6.793409255841564, 7.389443107728796, 7.843612614369808, 8.249931191305922, 8.696176439422418, 9.046438013783274, 9.749907717097335, 10.11085965322266, 10.59278525274928, 11.12164700595269, 11.48943927193593, 12.12814449684217, 12.43056602787048, 13.02933815842981
