| L(s) = 1 | + 0.470·3-s − 5-s − 2.71·7-s − 2.77·9-s + 5.55·11-s + 2.02·13-s − 0.470·15-s − 3.77·17-s − 19-s − 1.28·21-s + 5.77·23-s + 25-s − 2.71·27-s − 5.66·29-s − 7.55·31-s + 2.61·33-s + 2.71·35-s − 3.75·37-s + 0.954·39-s − 12.6·41-s + 9.43·43-s + 2.77·45-s − 11.1·47-s + 0.397·49-s − 1.77·51-s − 8.85·53-s − 5.55·55-s + ⋯ |
| L(s) = 1 | + 0.271·3-s − 0.447·5-s − 1.02·7-s − 0.926·9-s + 1.67·11-s + 0.562·13-s − 0.121·15-s − 0.916·17-s − 0.229·19-s − 0.279·21-s + 1.20·23-s + 0.200·25-s − 0.523·27-s − 1.05·29-s − 1.35·31-s + 0.455·33-s + 0.459·35-s − 0.616·37-s + 0.152·39-s − 1.97·41-s + 1.43·43-s + 0.414·45-s − 1.62·47-s + 0.0567·49-s − 0.249·51-s − 1.21·53-s − 0.749·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.470T + 3T^{2} \) |
| 7 | \( 1 + 2.71T + 7T^{2} \) |
| 11 | \( 1 - 5.55T + 11T^{2} \) |
| 13 | \( 1 - 2.02T + 13T^{2} \) |
| 17 | \( 1 + 3.77T + 17T^{2} \) |
| 23 | \( 1 - 5.77T + 23T^{2} \) |
| 29 | \( 1 + 5.66T + 29T^{2} \) |
| 31 | \( 1 + 7.55T + 31T^{2} \) |
| 37 | \( 1 + 3.75T + 37T^{2} \) |
| 41 | \( 1 + 12.6T + 41T^{2} \) |
| 43 | \( 1 - 9.43T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 + 8.85T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 9.45T + 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 - 4.94T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−9.077111960519254416733974828246, −8.540145357664334987504622043863, −7.35935278243352838393312261647, −6.57726309160933334728241837472, −6.03341621095946184504670611890, −4.78464883701679355126920616205, −3.61041505369148414840700250199, −3.25534928703236262790507389285, −1.72229109001735573255998072950, 0,
1.72229109001735573255998072950, 3.25534928703236262790507389285, 3.61041505369148414840700250199, 4.78464883701679355126920616205, 6.03341621095946184504670611890, 6.57726309160933334728241837472, 7.35935278243352838393312261647, 8.540145357664334987504622043863, 9.077111960519254416733974828246
