| L(s) = 1 | + 2·3-s + 2·5-s − 7-s + 9-s − 6·13-s + 4·15-s − 6·17-s + 4·19-s − 2·21-s − 25-s − 4·27-s + 4·29-s − 2·31-s − 2·35-s + 6·37-s − 12·39-s + 2·41-s + 4·43-s + 2·45-s + 10·47-s + 49-s − 12·51-s + 6·53-s + 8·57-s − 6·59-s + 14·61-s − 63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.66·13-s + 1.03·15-s − 1.45·17-s + 0.917·19-s − 0.436·21-s − 1/5·25-s − 0.769·27-s + 0.742·29-s − 0.359·31-s − 0.338·35-s + 0.986·37-s − 1.92·39-s + 0.312·41-s + 0.609·43-s + 0.298·45-s + 1.45·47-s + 1/7·49-s − 1.68·51-s + 0.824·53-s + 1.05·57-s − 0.781·59-s + 1.79·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.62191402700705, −14.10069415548366, −13.79988487500862, −13.15168236272289, −12.98329582973043, −12.13936200492807, −11.76298857841073, −11.02966277761860, −10.37639731606792, −9.863049896489346, −9.407364573706998, −9.148520527373143, −8.571766115900178, −7.842471512703408, −7.403938732075349, −6.881043388788419, −6.224683461923298, −5.579753766233467, −5.048724850149136, −4.297046276580944, −3.757062416739683, −2.844853534972201, −2.361235281768198, −2.236020135758187, −1.075884002338042, 0,
1.075884002338042, 2.236020135758187, 2.361235281768198, 2.844853534972201, 3.757062416739683, 4.297046276580944, 5.048724850149136, 5.579753766233467, 6.224683461923298, 6.881043388788419, 7.403938732075349, 7.842471512703408, 8.571766115900178, 9.148520527373143, 9.407364573706998, 9.863049896489346, 10.37639731606792, 11.02966277761860, 11.76298857841073, 12.13936200492807, 12.98329582973043, 13.15168236272289, 13.79988487500862, 14.10069415548366, 14.62191402700705
