L(s) = 1 | + 3-s − 5-s + 2·7-s + 9-s − 4·11-s − 15-s − 2·17-s − 19-s + 2·21-s + 2·23-s + 25-s + 27-s − 6·29-s − 4·33-s − 2·35-s − 8·37-s − 2·41-s + 6·43-s − 45-s + 2·47-s − 3·49-s − 2·51-s − 4·53-s + 4·55-s − 57-s − 4·59-s − 10·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.258·15-s − 0.485·17-s − 0.229·19-s + 0.436·21-s + 0.417·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.696·33-s − 0.338·35-s − 1.31·37-s − 0.312·41-s + 0.914·43-s − 0.149·45-s + 0.291·47-s − 3/7·49-s − 0.280·51-s − 0.549·53-s + 0.539·55-s − 0.132·57-s − 0.520·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−7.914500091996324325227939804167, −7.48887994829190695998310847673, −6.70144078901725447968994115359, −5.60239263108754950256317210277, −4.93517181743937497598214498755, −4.20972851668815285928885698978, −3.29736418184699888093706607156, −2.45098249104119650448755162716, −1.55512306337266666795172653462, 0,
1.55512306337266666795172653462, 2.45098249104119650448755162716, 3.29736418184699888093706607156, 4.20972851668815285928885698978, 4.93517181743937497598214498755, 5.60239263108754950256317210277, 6.70144078901725447968994115359, 7.48887994829190695998310847673, 7.914500091996324325227939804167