| L(s) = 1 | + 2-s + 0.347·3-s + 4-s + 2.61·5-s + 0.347·6-s − 2.07·7-s + 8-s − 2.87·9-s + 2.61·10-s − 5.99·11-s + 0.347·12-s − 4.11·13-s − 2.07·14-s + 0.906·15-s + 16-s − 0.962·17-s − 2.87·18-s − 1.56·19-s + 2.61·20-s − 0.722·21-s − 5.99·22-s + 6.37·23-s + 0.347·24-s + 1.81·25-s − 4.11·26-s − 2.04·27-s − 2.07·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.200·3-s + 0.5·4-s + 1.16·5-s + 0.141·6-s − 0.785·7-s + 0.353·8-s − 0.959·9-s + 0.825·10-s − 1.80·11-s + 0.100·12-s − 1.14·13-s − 0.555·14-s + 0.234·15-s + 0.250·16-s − 0.233·17-s − 0.678·18-s − 0.358·19-s + 0.583·20-s − 0.157·21-s − 1.27·22-s + 1.32·23-s + 0.0708·24-s + 0.363·25-s − 0.807·26-s − 0.392·27-s − 0.392·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 - 0.347T + 3T^{2} \) |
| 5 | \( 1 - 2.61T + 5T^{2} \) |
| 7 | \( 1 + 2.07T + 7T^{2} \) |
| 11 | \( 1 + 5.99T + 11T^{2} \) |
| 13 | \( 1 + 4.11T + 13T^{2} \) |
| 17 | \( 1 + 0.962T + 17T^{2} \) |
| 19 | \( 1 + 1.56T + 19T^{2} \) |
| 23 | \( 1 - 6.37T + 23T^{2} \) |
| 29 | \( 1 - 4.05T + 29T^{2} \) |
| 31 | \( 1 + 3.39T + 31T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 3.76T + 43T^{2} \) |
| 47 | \( 1 + 6.17T + 47T^{2} \) |
| 53 | \( 1 + 8.02T + 53T^{2} \) |
| 59 | \( 1 + 5.10T + 59T^{2} \) |
| 61 | \( 1 + 9.73T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 + 3.55T + 73T^{2} \) |
| 79 | \( 1 - 2.54T + 79T^{2} \) |
| 83 | \( 1 - 6.91T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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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
−8.412202642158458972135278990413, −7.60544967214771361555303726031, −6.70164006909423153632026597896, −6.05578833174930747238431217569, −5.16832447896701211190953884466, −4.92400456916099231014450731997, −3.25067305269864468553866460332, −2.74874264797591554549176637055, −2.03689801159647695536218701377, 0,
2.03689801159647695536218701377, 2.74874264797591554549176637055, 3.25067305269864468553866460332, 4.92400456916099231014450731997, 5.16832447896701211190953884466, 6.05578833174930747238431217569, 6.70164006909423153632026597896, 7.60544967214771361555303726031, 8.412202642158458972135278990413
