| L(s) = 1 | − 3-s + 9-s − 4·11-s + 2·13-s − 2·17-s + 4·23-s − 27-s − 2·29-s + 4·31-s + 4·33-s + 10·37-s − 2·39-s − 6·41-s + 4·43-s − 8·47-s + 2·51-s + 6·53-s + 12·59-s − 6·61-s + 12·67-s − 4·69-s + 6·73-s + 81-s − 12·83-s + 2·87-s − 6·89-s − 4·93-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.834·23-s − 0.192·27-s − 0.371·29-s + 0.718·31-s + 0.696·33-s + 1.64·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s − 1.16·47-s + 0.280·51-s + 0.824·53-s + 1.56·59-s − 0.768·61-s + 1.46·67-s − 0.481·69-s + 0.702·73-s + 1/9·81-s − 1.31·83-s + 0.214·87-s − 0.635·89-s − 0.414·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.396909666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.396909666\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 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
−15.29344201238856, −14.76173789234297, −14.02585833405359, −13.39325653487285, −13.01712683358370, −12.68359478890587, −11.86274354661968, −11.25718698242223, −11.05557219689025, −10.32863362305287, −9.888086281990083, −9.270130592118326, −8.487647966233456, −8.102771889431414, −7.418659462947448, −6.778311180812121, −6.288945382633013, −5.520237863741122, −5.146316816695638, −4.436081473865178, −3.795309129037511, −2.891877972786627, −2.348696601515008, −1.336022512397499, −0.4967991276573526,
0.4967991276573526, 1.336022512397499, 2.348696601515008, 2.891877972786627, 3.795309129037511, 4.436081473865178, 5.146316816695638, 5.520237863741122, 6.288945382633013, 6.778311180812121, 7.418659462947448, 8.102771889431414, 8.487647966233456, 9.270130592118326, 9.888086281990083, 10.32863362305287, 11.05557219689025, 11.25718698242223, 11.86274354661968, 12.68359478890587, 13.01712683358370, 13.39325653487285, 14.02585833405359, 14.76173789234297, 15.29344201238856
