| L(s) = 1 | + 3-s − 2·9-s + 6·11-s + 2·13-s − 6·17-s − 8·19-s − 3·23-s − 5·27-s + 3·29-s − 2·31-s + 6·33-s − 8·37-s + 2·39-s + 3·41-s − 5·43-s − 6·51-s − 12·53-s − 8·57-s + 61-s + 7·67-s − 3·69-s − 10·73-s − 4·79-s + 81-s + 3·83-s + 3·87-s + 3·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s + 1.80·11-s + 0.554·13-s − 1.45·17-s − 1.83·19-s − 0.625·23-s − 0.962·27-s + 0.557·29-s − 0.359·31-s + 1.04·33-s − 1.31·37-s + 0.320·39-s + 0.468·41-s − 0.762·43-s − 0.840·51-s − 1.64·53-s − 1.05·57-s + 0.128·61-s + 0.855·67-s − 0.361·69-s − 1.17·73-s − 0.450·79-s + 1/9·81-s + 0.329·83-s + 0.321·87-s + 0.317·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.211270859347756885349671452319, −7.04879334883551896766303355152, −6.40678059519075227049276802788, −6.03236251086300149239737522411, −4.73220950631722382800727147838, −4.04233006113911691325892171640, −3.43864198876714594107947372560, −2.30245811067245526829398566091, −1.60712841656694145068894267650, 0,
1.60712841656694145068894267650, 2.30245811067245526829398566091, 3.43864198876714594107947372560, 4.04233006113911691325892171640, 4.73220950631722382800727147838, 6.03236251086300149239737522411, 6.40678059519075227049276802788, 7.04879334883551896766303355152, 8.211270859347756885349671452319
