| L(s) = 1 | + 3-s + 4·5-s − 2·7-s + 9-s − 4·11-s + 4·15-s + 2·17-s − 2·19-s − 2·21-s + 11·25-s + 27-s − 6·29-s − 10·31-s − 4·33-s − 8·35-s − 10·37-s − 8·41-s − 4·43-s + 4·45-s − 4·47-s − 3·49-s + 2·51-s − 10·53-s − 16·55-s − 2·57-s − 8·59-s − 14·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s − 0.755·7-s + 1/3·9-s − 1.20·11-s + 1.03·15-s + 0.485·17-s − 0.458·19-s − 0.436·21-s + 11/5·25-s + 0.192·27-s − 1.11·29-s − 1.79·31-s − 0.696·33-s − 1.35·35-s − 1.64·37-s − 1.24·41-s − 0.609·43-s + 0.596·45-s − 0.583·47-s − 3/7·49-s + 0.280·51-s − 1.37·53-s − 2.15·55-s − 0.264·57-s − 1.04·59-s − 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.49494212843899544246880444280, −6.63917471623644010166347697023, −6.18799167779584509086170063751, −5.21478505552964830095770134421, −5.09137955508891753871660568663, −3.53356351432599386462319862398, −3.11302755763053880426051678408, −2.05801550286862140543150513505, −1.72378093788660960790427307428, 0,
1.72378093788660960790427307428, 2.05801550286862140543150513505, 3.11302755763053880426051678408, 3.53356351432599386462319862398, 5.09137955508891753871660568663, 5.21478505552964830095770134421, 6.18799167779584509086170063751, 6.63917471623644010166347697023, 7.49494212843899544246880444280
