| L(s) = 1 | + 3-s − 7-s + 9-s − 4.77·11-s − 13-s − 19-s − 21-s + 23-s + 27-s + 2.77·29-s + 5.77·31-s − 4.77·33-s + 10·37-s − 39-s + 6.77·41-s − 10.5·43-s − 3.54·47-s − 6·49-s − 4·53-s − 57-s + 11.5·59-s − 11.3·61-s − 63-s + 1.77·67-s + 69-s + 4·71-s + 16.7·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 0.333·9-s − 1.43·11-s − 0.277·13-s − 0.229·19-s − 0.218·21-s + 0.208·23-s + 0.192·27-s + 0.514·29-s + 1.03·31-s − 0.830·33-s + 1.64·37-s − 0.160·39-s + 1.05·41-s − 1.60·43-s − 0.516·47-s − 0.857·49-s − 0.549·53-s − 0.132·57-s + 1.50·59-s − 1.44·61-s − 0.125·63-s + 0.216·67-s + 0.120·69-s + 0.474·71-s + 1.96·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.924826658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.924826658\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 + 4.77T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 29 | \( 1 - 2.77T + 29T^{2} \) |
| 31 | \( 1 - 5.77T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 - 6.77T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 3.54T + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 - 1.77T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 16.7T + 73T^{2} \) |
| 79 | \( 1 + 16.3T + 79T^{2} \) |
| 83 | \( 1 - 16.7T + 83T^{2} \) |
| 89 | \( 1 + 7.54T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−8.135780389345467938777799021591, −7.37518346022441379262096801407, −6.59918638872160392140975424060, −5.92076269393380916663356440708, −4.96007721565057716282559603843, −4.48120674875044393714458985433, −3.35066577347562365778164022534, −2.77971930060395436306962027568, −2.03744861198898925437770973351, −0.66469321296268754096452268005,
0.66469321296268754096452268005, 2.03744861198898925437770973351, 2.77971930060395436306962027568, 3.35066577347562365778164022534, 4.48120674875044393714458985433, 4.96007721565057716282559603843, 5.92076269393380916663356440708, 6.59918638872160392140975424060, 7.37518346022441379262096801407, 8.135780389345467938777799021591
