| L(s) = 1 | − 3-s + 9-s − 3·13-s + 2·17-s + 19-s + 2·23-s − 27-s − 8·29-s − 8·31-s − 7·37-s + 3·39-s − 8·43-s + 10·47-s − 2·51-s + 14·53-s − 57-s + 10·59-s − 7·61-s − 5·67-s − 2·69-s + 12·71-s − 11·73-s + 7·79-s + 81-s − 14·83-s + 8·87-s + 6·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.832·13-s + 0.485·17-s + 0.229·19-s + 0.417·23-s − 0.192·27-s − 1.48·29-s − 1.43·31-s − 1.15·37-s + 0.480·39-s − 1.21·43-s + 1.45·47-s − 0.280·51-s + 1.92·53-s − 0.132·57-s + 1.30·59-s − 0.896·61-s − 0.610·67-s − 0.240·69-s + 1.42·71-s − 1.28·73-s + 0.787·79-s + 1/9·81-s − 1.53·83-s + 0.857·87-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−14.73095881869097, −14.11010359047179, −13.48563855680084, −13.01073147544035, −12.55784375885712, −11.95606460752729, −11.66460038622333, −11.03556040787214, −10.47582464103819, −10.10341614037798, −9.474009790933225, −8.969359411998086, −8.468400622195011, −7.530938947503383, −7.304054301098625, −6.877608487984404, −5.988369322020594, −5.483497737244619, −5.178940100921644, −4.436608359285816, −3.712615342902426, −3.269312523620163, −2.270305694903136, −1.775759372634011, −0.8171657289212251, 0,
0.8171657289212251, 1.775759372634011, 2.270305694903136, 3.269312523620163, 3.712615342902426, 4.436608359285816, 5.178940100921644, 5.483497737244619, 5.988369322020594, 6.877608487984404, 7.304054301098625, 7.530938947503383, 8.468400622195011, 8.969359411998086, 9.474009790933225, 10.10341614037798, 10.47582464103819, 11.03556040787214, 11.66460038622333, 11.95606460752729, 12.55784375885712, 13.01073147544035, 13.48563855680084, 14.11010359047179, 14.73095881869097
