| L(s) = 1 | + 2-s + 4-s + 4·7-s + 8-s + 4·13-s + 4·14-s + 16-s + 6·17-s + 19-s − 6·23-s + 4·26-s + 4·28-s − 6·29-s + 2·31-s + 32-s + 6·34-s + 4·37-s + 38-s − 6·41-s + 4·43-s − 6·46-s + 6·47-s + 9·49-s + 4·52-s + 6·53-s + 4·56-s − 6·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.51·7-s + 0.353·8-s + 1.10·13-s + 1.06·14-s + 1/4·16-s + 1.45·17-s + 0.229·19-s − 1.25·23-s + 0.784·26-s + 0.755·28-s − 1.11·29-s + 0.359·31-s + 0.176·32-s + 1.02·34-s + 0.657·37-s + 0.162·38-s − 0.937·41-s + 0.609·43-s − 0.884·46-s + 0.875·47-s + 9/7·49-s + 0.554·52-s + 0.824·53-s + 0.534·56-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.780728982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.780728982\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−7.70097791479153211453619417755, −7.22584614781181397611801863374, −6.12602123734563039032805201410, −5.61440500425845973312912145894, −5.11806274676663495258347375100, −4.09030165888843990239173890258, −3.79449350274307596703126602160, −2.67810585413014879641600772416, −1.74961074166436521935542159057, −1.06259828383013969636779705746,
1.06259828383013969636779705746, 1.74961074166436521935542159057, 2.67810585413014879641600772416, 3.79449350274307596703126602160, 4.09030165888843990239173890258, 5.11806274676663495258347375100, 5.61440500425845973312912145894, 6.12602123734563039032805201410, 7.22584614781181397611801863374, 7.70097791479153211453619417755
