| L(s) = 1 | − 2-s + 4-s + 3.53·5-s + 3.71·7-s − 8-s − 3.53·10-s + 5.29·11-s − 0.226·13-s − 3.71·14-s + 16-s + 1.65·17-s + 3.53·20-s − 5.29·22-s − 8.68·23-s + 7.47·25-s + 0.226·26-s + 3.71·28-s − 0.120·29-s + 3.12·31-s − 32-s − 1.65·34-s + 13.1·35-s + 5.12·37-s − 3.53·40-s − 7.10·41-s − 5.35·43-s + 5.29·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.57·5-s + 1.40·7-s − 0.353·8-s − 1.11·10-s + 1.59·11-s − 0.0628·13-s − 0.993·14-s + 0.250·16-s + 0.400·17-s + 0.789·20-s − 1.12·22-s − 1.80·23-s + 1.49·25-s + 0.0444·26-s + 0.702·28-s − 0.0223·29-s + 0.560·31-s − 0.176·32-s − 0.283·34-s + 2.21·35-s + 0.843·37-s − 0.558·40-s − 1.10·41-s − 0.816·43-s + 0.797·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.824437444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.824437444\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 3.53T + 5T^{2} \) |
| 7 | \( 1 - 3.71T + 7T^{2} \) |
| 11 | \( 1 - 5.29T + 11T^{2} \) |
| 13 | \( 1 + 0.226T + 13T^{2} \) |
| 17 | \( 1 - 1.65T + 17T^{2} \) |
| 23 | \( 1 + 8.68T + 23T^{2} \) |
| 29 | \( 1 + 0.120T + 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 37 | \( 1 - 5.12T + 37T^{2} \) |
| 41 | \( 1 + 7.10T + 41T^{2} \) |
| 43 | \( 1 + 5.35T + 43T^{2} \) |
| 47 | \( 1 - 2.50T + 47T^{2} \) |
| 53 | \( 1 + 5.93T + 53T^{2} \) |
| 59 | \( 1 + 0.218T + 59T^{2} \) |
| 61 | \( 1 + 1.57T + 61T^{2} \) |
| 67 | \( 1 - 15.4T + 67T^{2} \) |
| 71 | \( 1 - 1.35T + 71T^{2} \) |
| 73 | \( 1 - 2.42T + 73T^{2} \) |
| 79 | \( 1 + 2.86T + 79T^{2} \) |
| 83 | \( 1 + 1.92T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−8.229861464932672787251598786033, −7.38767749456648496284681375307, −6.45034015356447766866617478881, −6.12073034554759312695997340078, −5.28739471237764673306215887515, −4.51995225010620664373753252712, −3.54231658900196952027362594348, −2.23223999119825592970115224311, −1.76112205255512231219806819376, −1.07562062662522147592659764116,
1.07562062662522147592659764116, 1.76112205255512231219806819376, 2.23223999119825592970115224311, 3.54231658900196952027362594348, 4.51995225010620664373753252712, 5.28739471237764673306215887515, 6.12073034554759312695997340078, 6.45034015356447766866617478881, 7.38767749456648496284681375307, 8.229861464932672787251598786033
