| L(s) = 1 | + 2·3-s − 2·4-s − 5-s − 7-s + 9-s − 11-s − 4·12-s − 13-s − 2·15-s + 4·16-s + 2·17-s − 19-s + 2·20-s − 2·21-s − 5·23-s − 4·25-s − 4·27-s + 2·28-s − 5·29-s − 9·31-s − 2·33-s + 35-s − 2·36-s + 10·37-s − 2·39-s − 2·41-s − 13·43-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.15·12-s − 0.277·13-s − 0.516·15-s + 16-s + 0.485·17-s − 0.229·19-s + 0.447·20-s − 0.436·21-s − 1.04·23-s − 4/5·25-s − 0.769·27-s + 0.377·28-s − 0.928·29-s − 1.61·31-s − 0.348·33-s + 0.169·35-s − 1/3·36-s + 1.64·37-s − 0.320·39-s − 0.312·41-s − 1.98·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−9.640083292373987058603191049812, −8.650195885178034734764412957721, −8.025597150713170782148421117427, −7.46595058593106128092987481128, −6.03687681304360526387090452910, −5.06188428719186619231283594875, −3.85872420740283031925332467940, −3.41704386442178853478657445265, −2.05343142591405820873295205760, 0,
2.05343142591405820873295205760, 3.41704386442178853478657445265, 3.85872420740283031925332467940, 5.06188428719186619231283594875, 6.03687681304360526387090452910, 7.46595058593106128092987481128, 8.025597150713170782148421117427, 8.650195885178034734764412957721, 9.640083292373987058603191049812
