| L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s − 4·13-s + 15-s − 2·17-s + 19-s − 2·21-s + 6·23-s + 25-s − 27-s − 2·29-s − 2·35-s − 4·37-s + 4·39-s + 2·41-s − 2·43-s − 45-s + 6·47-s − 3·49-s + 2·51-s − 12·53-s − 57-s + 4·59-s + 6·61-s + 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.10·13-s + 0.258·15-s − 0.485·17-s + 0.229·19-s − 0.436·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.338·35-s − 0.657·37-s + 0.640·39-s + 0.312·41-s − 0.304·43-s − 0.149·45-s + 0.875·47-s − 3/7·49-s + 0.280·51-s − 1.64·53-s − 0.132·57-s + 0.520·59-s + 0.768·61-s + 0.251·63-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 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.81069973683552537764219633572, −7.26638470980080213201722549918, −6.64071348241599421036541597605, −5.62078747703675619671923600515, −4.91258465380057120474005063394, −4.47416950603366237031202564520, −3.39120538263897498986822023931, −2.37176213738933915431121152151, −1.28629106315012200211845850675, 0,
1.28629106315012200211845850675, 2.37176213738933915431121152151, 3.39120538263897498986822023931, 4.47416950603366237031202564520, 4.91258465380057120474005063394, 5.62078747703675619671923600515, 6.64071348241599421036541597605, 7.26638470980080213201722549918, 7.81069973683552537764219633572
