| L(s) = 1 | − 3-s − 2·5-s − 7-s + 9-s − 2·13-s + 2·15-s + 2·17-s + 19-s + 21-s − 4·23-s − 25-s − 27-s + 2·29-s + 2·35-s − 10·37-s + 2·39-s − 10·41-s + 4·43-s − 2·45-s + 12·47-s + 49-s − 2·51-s − 6·53-s − 57-s − 4·59-s + 14·61-s − 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s − 0.554·13-s + 0.516·15-s + 0.485·17-s + 0.229·19-s + 0.218·21-s − 0.834·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.338·35-s − 1.64·37-s + 0.320·39-s − 1.56·41-s + 0.609·43-s − 0.298·45-s + 1.75·47-s + 1/7·49-s − 0.280·51-s − 0.824·53-s − 0.132·57-s − 0.520·59-s + 1.79·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8019029799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8019029799\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−8.591695184832218085816253731280, −7.79079379173587089245475615940, −7.22754363342506134049220656124, −6.45615318567160936334373690750, −5.59606026598333908724694165234, −4.85150909211420773997211690459, −3.94502182779103014780841209433, −3.26682865336112103968464934285, −1.98697251530591629789599134724, −0.53972525243018190332018985759,
0.53972525243018190332018985759, 1.98697251530591629789599134724, 3.26682865336112103968464934285, 3.94502182779103014780841209433, 4.85150909211420773997211690459, 5.59606026598333908724694165234, 6.45615318567160936334373690750, 7.22754363342506134049220656124, 7.79079379173587089245475615940, 8.591695184832218085816253731280
