| L(s) = 1 | + 2-s − 2·3-s + 4-s − 5-s − 2·6-s + 7-s + 8-s + 9-s − 10-s − 3·11-s − 2·12-s − 4·13-s + 14-s + 2·15-s + 16-s + 4·17-s + 18-s − 2·19-s − 20-s − 2·21-s − 3·22-s − 8·23-s − 2·24-s + 25-s − 4·26-s + 4·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.904·11-s − 0.577·12-s − 1.10·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.458·19-s − 0.223·20-s − 0.436·21-s − 0.639·22-s − 1.66·23-s − 0.408·24-s + 1/5·25-s − 0.784·26-s + 0.769·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−10.46898806314781024415089882103, −9.437456289893734658076637002243, −7.85228283712943664674539258094, −7.49195055843111758974760144580, −6.15372064441272782436251437701, −5.46048792361006441510673532800, −4.76552753630307286926944563151, −3.61886614257202043335030200992, −2.13061104084151768873353401945, 0,
2.13061104084151768873353401945, 3.61886614257202043335030200992, 4.76552753630307286926944563151, 5.46048792361006441510673532800, 6.15372064441272782436251437701, 7.49195055843111758974760144580, 7.85228283712943664674539258094, 9.437456289893734658076637002243, 10.46898806314781024415089882103
