| L(s) = 1 | + 2-s + 4-s + 2·7-s + 8-s + 2·13-s + 2·14-s + 16-s + 17-s − 4·19-s + 6·23-s − 5·25-s + 2·26-s + 2·28-s − 10·31-s + 32-s + 34-s + 8·37-s − 4·38-s − 6·41-s − 4·43-s + 6·46-s − 12·47-s − 3·49-s − 5·50-s + 2·52-s − 6·53-s + 2·56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s + 0.554·13-s + 0.534·14-s + 1/4·16-s + 0.242·17-s − 0.917·19-s + 1.25·23-s − 25-s + 0.392·26-s + 0.377·28-s − 1.79·31-s + 0.176·32-s + 0.171·34-s + 1.31·37-s − 0.648·38-s − 0.937·41-s − 0.609·43-s + 0.884·46-s − 1.75·47-s − 3/7·49-s − 0.707·50-s + 0.277·52-s − 0.824·53-s + 0.267·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 306 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 306 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.068306077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.068306077\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−11.53385169968493935461100938953, −11.14807860206782462084621558636, −10.00723343046726241119720950602, −8.745924693621362183100177299272, −7.81014283472581039979750102513, −6.70932306148654350305246926067, −5.60715979666329449780909401068, −4.60759445774375135200288872743, −3.42809860399723990451499242436, −1.81193945201237649695550582777,
1.81193945201237649695550582777, 3.42809860399723990451499242436, 4.60759445774375135200288872743, 5.60715979666329449780909401068, 6.70932306148654350305246926067, 7.81014283472581039979750102513, 8.745924693621362183100177299272, 10.00723343046726241119720950602, 11.14807860206782462084621558636, 11.53385169968493935461100938953
