| L(s) = 1 | − 5.06·2-s − 3·3-s + 17.6·4-s + 12.0·5-s + 15.1·6-s − 27.2·7-s − 48.7·8-s + 9·9-s − 61.2·10-s − 47.0·11-s − 52.8·12-s + 43.5·13-s + 137.·14-s − 36.2·15-s + 105.·16-s + 7.57·17-s − 45.5·18-s − 35.0·19-s + 213.·20-s + 81.6·21-s + 238.·22-s + 20.2·23-s + 146.·24-s + 21.2·25-s − 220.·26-s − 27·27-s − 479.·28-s + ⋯ |
| L(s) = 1 | − 1.78·2-s − 0.577·3-s + 2.20·4-s + 1.08·5-s + 1.03·6-s − 1.46·7-s − 2.15·8-s + 0.333·9-s − 1.93·10-s − 1.29·11-s − 1.27·12-s + 0.928·13-s + 2.62·14-s − 0.624·15-s + 1.65·16-s + 0.108·17-s − 0.596·18-s − 0.422·19-s + 2.38·20-s + 0.848·21-s + 2.30·22-s + 0.184·23-s + 1.24·24-s + 0.170·25-s − 1.66·26-s − 0.192·27-s − 3.23·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4737477439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4737477439\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 157 | \( 1 + 157T \) |
| good | 2 | \( 1 + 5.06T + 8T^{2} \) |
| 5 | \( 1 - 12.0T + 125T^{2} \) |
| 7 | \( 1 + 27.2T + 343T^{2} \) |
| 11 | \( 1 + 47.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 43.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 7.57T + 4.91e3T^{2} \) |
| 19 | \( 1 + 35.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 20.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 267.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 67.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 274.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 94.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 504.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 70.0T + 1.03e5T^{2} \) |
| 53 | \( 1 + 82.4T + 1.48e5T^{2} \) |
| 59 | \( 1 - 625.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 4.48T + 2.26e5T^{2} \) |
| 67 | \( 1 + 193.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 304.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 689.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.17e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 379.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 241.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.07e3T + 9.12e5T^{2} \) |
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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
−10.19829944105531040628968460817, −9.966639912567817387058884343476, −8.958879362744539410496264412719, −8.139160913590468092854690312567, −6.80912278123982973164795277093, −6.38567235758135150292892426033, −5.38247085454843822385565015628, −3.14750054995310666985855249977, −1.97326434729862792663036823020, −0.56110244236875198319347078901,
0.56110244236875198319347078901, 1.97326434729862792663036823020, 3.14750054995310666985855249977, 5.38247085454843822385565015628, 6.38567235758135150292892426033, 6.80912278123982973164795277093, 8.139160913590468092854690312567, 8.958879362744539410496264412719, 9.966639912567817387058884343476, 10.19829944105531040628968460817
