| L(s) = 1 | − 2-s + 4-s − 2.44·5-s − 8-s + 2.44·10-s + 2.44·11-s − 2·13-s + 16-s − 4.89·17-s − 7.34·19-s − 2.44·20-s − 2.44·22-s + 0.999·25-s + 2·26-s + 6·29-s − 2·31-s − 32-s + 4.89·34-s − 7.34·37-s + 7.34·38-s + 2.44·40-s + 6·41-s − 7.34·43-s + 2.44·44-s + 6·47-s − 7·49-s − 0.999·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.09·5-s − 0.353·8-s + 0.774·10-s + 0.738·11-s − 0.554·13-s + 0.250·16-s − 1.18·17-s − 1.68·19-s − 0.547·20-s − 0.522·22-s + 0.199·25-s + 0.392·26-s + 1.11·29-s − 0.359·31-s − 0.176·32-s + 0.840·34-s − 1.20·37-s + 1.19·38-s + 0.387·40-s + 0.937·41-s − 1.12·43-s + 0.369·44-s + 0.875·47-s − 49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4828382720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4828382720\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 2.44T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 2.44T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 + 7.34T + 19T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 7.34T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 7.34T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 2.44T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 7.34T + 61T^{2} \) |
| 67 | \( 1 - 7.34T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 7.34T + 83T^{2} \) |
| 89 | \( 1 - 9.79T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−7.82800857614988651980638298009, −6.91748159902277717884811929242, −6.71371382771980118802536478165, −5.84396227241349729090651034664, −4.66792264429873066365219816207, −4.25494128827585354924611370968, −3.43965408762515013325109926408, −2.47433656631125122832731118822, −1.66471839636823817707986107990, −0.36131266297998759267203804154,
0.36131266297998759267203804154, 1.66471839636823817707986107990, 2.47433656631125122832731118822, 3.43965408762515013325109926408, 4.25494128827585354924611370968, 4.66792264429873066365219816207, 5.84396227241349729090651034664, 6.71371382771980118802536478165, 6.91748159902277717884811929242, 7.82800857614988651980638298009
