| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s + 9-s − 10-s − 12-s − 13-s − 14-s + 15-s + 16-s + 2·17-s + 18-s − 4·19-s − 20-s + 21-s + 4·23-s − 24-s + 25-s − 26-s − 27-s − 28-s + 6·29-s + 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.218·21-s + 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + 1.11·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 330330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.77205936151195, −12.28311952506792, −11.95813021341786, −11.75830774348799, −10.82037659147041, −10.77665509433194, −10.34741313331828, −9.682681843873763, −9.277654998742069, −8.550937395902311, −8.275746547784695, −7.543214338976647, −7.163295386783551, −6.723468338647070, −6.286539609957059, −5.697799515020432, −5.397206659239778, −4.611837589060089, −4.406493724345157, −3.894267982237830, −3.169929960494590, −2.768530308464190, −2.212740164037910, −1.317658092853030, −0.8118118533581349, 0,
0.8118118533581349, 1.317658092853030, 2.212740164037910, 2.768530308464190, 3.169929960494590, 3.894267982237830, 4.406493724345157, 4.611837589060089, 5.397206659239778, 5.697799515020432, 6.286539609957059, 6.723468338647070, 7.163295386783551, 7.543214338976647, 8.275746547784695, 8.550937395902311, 9.277654998742069, 9.682681843873763, 10.34741313331828, 10.77665509433194, 10.82037659147041, 11.75830774348799, 11.95813021341786, 12.28311952506792, 12.77205936151195
