| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 13-s − 15-s − 2·17-s − 4·19-s − 21-s + 4·23-s + 25-s − 27-s − 2·29-s + 35-s − 6·37-s + 39-s + 10·41-s − 8·43-s + 45-s + 12·47-s + 49-s + 2·51-s − 10·53-s + 4·57-s + 4·59-s + 2·61-s + 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.277·13-s − 0.258·15-s − 0.485·17-s − 0.917·19-s − 0.218·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.169·35-s − 0.986·37-s + 0.160·39-s + 1.56·41-s − 1.21·43-s + 0.149·45-s + 1.75·47-s + 1/7·49-s + 0.280·51-s − 1.37·53-s + 0.529·57-s + 0.520·59-s + 0.256·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−17.03493257256426, −16.29824989599626, −15.64301715362273, −15.11815314766361, −14.55540170871609, −13.92812577091461, −13.32736168825922, −12.68036821636668, −12.32762998739794, −11.47978301839225, −10.95591604278724, −10.57375330547079, −9.810007547232107, −9.191908684895011, −8.598594584061831, −7.881710960451812, −7.042932612498742, −6.656244452064868, −5.805418194245614, −5.299820547802966, −4.559382721834540, −3.957778585054400, −2.848779703385226, −2.071716128220198, −1.207754963458607, 0,
1.207754963458607, 2.071716128220198, 2.848779703385226, 3.957778585054400, 4.559382721834540, 5.299820547802966, 5.805418194245614, 6.656244452064868, 7.042932612498742, 7.881710960451812, 8.598594584061831, 9.191908684895011, 9.810007547232107, 10.57375330547079, 10.95591604278724, 11.47978301839225, 12.32762998739794, 12.68036821636668, 13.32736168825922, 13.92812577091461, 14.55540170871609, 15.11815314766361, 15.64301715362273, 16.29824989599626, 17.03493257256426
