| L(s) = 1 | + 3-s + 4·7-s − 2·9-s − 11-s + 2·13-s + 2·19-s + 4·21-s + 9·23-s − 5·27-s + 4·29-s − 5·31-s − 33-s + 9·37-s + 2·39-s + 2·41-s − 6·43-s − 4·47-s + 9·49-s + 6·53-s + 2·57-s + 5·59-s − 8·63-s − 13·67-s + 9·69-s + 71-s − 14·73-s − 4·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s − 2/3·9-s − 0.301·11-s + 0.554·13-s + 0.458·19-s + 0.872·21-s + 1.87·23-s − 0.962·27-s + 0.742·29-s − 0.898·31-s − 0.174·33-s + 1.47·37-s + 0.320·39-s + 0.312·41-s − 0.914·43-s − 0.583·47-s + 9/7·49-s + 0.824·53-s + 0.264·57-s + 0.650·59-s − 1.00·63-s − 1.58·67-s + 1.08·69-s + 0.118·71-s − 1.63·73-s − 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.232052473\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.232052473\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−7.77173300291814190695834897021, −7.35803936263715590141987954660, −6.38101157222320412649214341409, −5.52149830778961358957909895997, −5.01858770943667581245859835212, −4.30312075205520165788251158272, −3.31744081581445262537376748155, −2.68992511387026479126151876075, −1.77309155415494072493662094200, −0.889538454615368568189074247395,
0.889538454615368568189074247395, 1.77309155415494072493662094200, 2.68992511387026479126151876075, 3.31744081581445262537376748155, 4.30312075205520165788251158272, 5.01858770943667581245859835212, 5.52149830778961358957909895997, 6.38101157222320412649214341409, 7.35803936263715590141987954660, 7.77173300291814190695834897021
