| L(s) = 1 | + 2-s − 2.44·3-s + 4-s − 2.44·6-s + 8-s + 2.99·9-s − 4.89·11-s − 2.44·12-s + 4.44·13-s + 16-s + 2·17-s + 2.99·18-s − 1.55·19-s − 4.89·22-s − 2.89·23-s − 2.44·24-s + 4.44·26-s + 6.89·29-s − 8.89·31-s + 32-s + 11.9·33-s + 2·34-s + 2.99·36-s − 2·37-s − 1.55·38-s − 10.8·39-s + 1.10·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.41·3-s + 0.5·4-s − 0.999·6-s + 0.353·8-s + 0.999·9-s − 1.47·11-s − 0.707·12-s + 1.23·13-s + 0.250·16-s + 0.485·17-s + 0.707·18-s − 0.355·19-s − 1.04·22-s − 0.604·23-s − 0.499·24-s + 0.872·26-s + 1.28·29-s − 1.59·31-s + 0.176·32-s + 2.08·33-s + 0.342·34-s + 0.499·36-s − 0.328·37-s − 0.251·38-s − 1.74·39-s + 0.171·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.502677631\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.502677631\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.44T + 3T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 - 4.44T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 1.55T + 19T^{2} \) |
| 23 | \( 1 + 2.89T + 23T^{2} \) |
| 29 | \( 1 - 6.89T + 29T^{2} \) |
| 31 | \( 1 + 8.89T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 1.10T + 41T^{2} \) |
| 43 | \( 1 - 0.898T + 43T^{2} \) |
| 47 | \( 1 - 8.89T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 1.55T + 59T^{2} \) |
| 61 | \( 1 + 3.55T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 1.10T + 71T^{2} \) |
| 73 | \( 1 - 2.89T + 73T^{2} \) |
| 79 | \( 1 - 6.89T + 79T^{2} \) |
| 83 | \( 1 + 2.44T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−8.854904646512019885680717432001, −7.993628104634749638104467263975, −7.17628674278209051487346451917, −6.31475450720530471303113856378, −5.68805730434565079173099842072, −5.23708276580679400235613009186, −4.32725631917608889902117991695, −3.37939618012230527233218531033, −2.17227626850892447233460060514, −0.74752741983909619393551071916,
0.74752741983909619393551071916, 2.17227626850892447233460060514, 3.37939618012230527233218531033, 4.32725631917608889902117991695, 5.23708276580679400235613009186, 5.68805730434565079173099842072, 6.31475450720530471303113856378, 7.17628674278209051487346451917, 7.993628104634749638104467263975, 8.854904646512019885680717432001
