| L(s) = 1 | + 2-s + 2.78·3-s + 4-s + 0.221·5-s + 2.78·6-s + 0.625·7-s + 8-s + 4.73·9-s + 0.221·10-s + 3.00·11-s + 2.78·12-s − 1.42·13-s + 0.625·14-s + 0.615·15-s + 16-s + 4.60·17-s + 4.73·18-s − 6.76·19-s + 0.221·20-s + 1.73·21-s + 3.00·22-s + 1.59·23-s + 2.78·24-s − 4.95·25-s − 1.42·26-s + 4.82·27-s + 0.625·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.60·3-s + 0.5·4-s + 0.0989·5-s + 1.13·6-s + 0.236·7-s + 0.353·8-s + 1.57·9-s + 0.0699·10-s + 0.905·11-s + 0.802·12-s − 0.396·13-s + 0.167·14-s + 0.158·15-s + 0.250·16-s + 1.11·17-s + 1.11·18-s − 1.55·19-s + 0.0494·20-s + 0.379·21-s + 0.640·22-s + 0.331·23-s + 0.567·24-s − 0.990·25-s − 0.280·26-s + 0.928·27-s + 0.118·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.797122839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.797122839\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 751 | \( 1 + T \) |
| good | 3 | \( 1 - 2.78T + 3T^{2} \) |
| 5 | \( 1 - 0.221T + 5T^{2} \) |
| 7 | \( 1 - 0.625T + 7T^{2} \) |
| 11 | \( 1 - 3.00T + 11T^{2} \) |
| 13 | \( 1 + 1.42T + 13T^{2} \) |
| 17 | \( 1 - 4.60T + 17T^{2} \) |
| 19 | \( 1 + 6.76T + 19T^{2} \) |
| 23 | \( 1 - 1.59T + 23T^{2} \) |
| 29 | \( 1 + 7.25T + 29T^{2} \) |
| 31 | \( 1 + 6.25T + 31T^{2} \) |
| 37 | \( 1 - 2.62T + 37T^{2} \) |
| 41 | \( 1 + 6.82T + 41T^{2} \) |
| 43 | \( 1 - 2.62T + 43T^{2} \) |
| 47 | \( 1 - 8.53T + 47T^{2} \) |
| 53 | \( 1 - 4.50T + 53T^{2} \) |
| 59 | \( 1 - 1.39T + 59T^{2} \) |
| 61 | \( 1 - 9.06T + 61T^{2} \) |
| 67 | \( 1 + 15.7T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 - 6.05T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 + 16.7T + 97T^{2} \) |
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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
−9.360077835373229617386481000147, −8.688027373931715987112424218113, −7.82489227300709627441073705358, −7.24865615236471529035881804713, −6.24074083441269329159000669377, −5.21205887896312236374362437921, −3.97280695816152638183541812921, −3.65321986940687718386584472606, −2.42966694850068378682801503744, −1.68703629518440142585532702667,
1.68703629518440142585532702667, 2.42966694850068378682801503744, 3.65321986940687718386584472606, 3.97280695816152638183541812921, 5.21205887896312236374362437921, 6.24074083441269329159000669377, 7.24865615236471529035881804713, 7.82489227300709627441073705358, 8.688027373931715987112424218113, 9.360077835373229617386481000147
