| L(s) = 1 | − 0.779·2-s − 1.24·3-s − 1.39·4-s + 1.96·5-s + 0.973·6-s + 3.90·7-s + 2.64·8-s − 1.43·9-s − 1.52·10-s + 5.25·11-s + 1.73·12-s + 13-s − 3.04·14-s − 2.45·15-s + 0.722·16-s − 0.637·17-s + 1.12·18-s + 6.18·19-s − 2.73·20-s − 4.87·21-s − 4.09·22-s − 2.06·23-s − 3.30·24-s − 1.14·25-s − 0.779·26-s + 5.54·27-s − 5.43·28-s + ⋯ |
| L(s) = 1 | − 0.551·2-s − 0.721·3-s − 0.696·4-s + 0.877·5-s + 0.397·6-s + 1.47·7-s + 0.934·8-s − 0.479·9-s − 0.483·10-s + 1.58·11-s + 0.502·12-s + 0.277·13-s − 0.813·14-s − 0.632·15-s + 0.180·16-s − 0.154·17-s + 0.264·18-s + 1.41·19-s − 0.610·20-s − 1.06·21-s − 0.873·22-s − 0.429·23-s − 0.674·24-s − 0.229·25-s − 0.152·26-s + 1.06·27-s − 1.02·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.236551642\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.236551642\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 0.779T + 2T^{2} \) |
| 3 | \( 1 + 1.24T + 3T^{2} \) |
| 5 | \( 1 - 1.96T + 5T^{2} \) |
| 7 | \( 1 - 3.90T + 7T^{2} \) |
| 11 | \( 1 - 5.25T + 11T^{2} \) |
| 17 | \( 1 + 0.637T + 17T^{2} \) |
| 19 | \( 1 - 6.18T + 19T^{2} \) |
| 23 | \( 1 + 2.06T + 23T^{2} \) |
| 29 | \( 1 + 2.34T + 29T^{2} \) |
| 31 | \( 1 - 0.549T + 31T^{2} \) |
| 37 | \( 1 + 7.82T + 37T^{2} \) |
| 41 | \( 1 - 7.09T + 41T^{2} \) |
| 43 | \( 1 + 5.70T + 43T^{2} \) |
| 47 | \( 1 - 4.06T + 47T^{2} \) |
| 53 | \( 1 - 7.88T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 9.82T + 61T^{2} \) |
| 67 | \( 1 - 9.54T + 67T^{2} \) |
| 71 | \( 1 - 1.65T + 71T^{2} \) |
| 73 | \( 1 - 8.52T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 + 9.50T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−9.465876405028484689637476517904, −8.946785705386854011519557198208, −8.177721846180402994333234296894, −7.27764101552667293667112897119, −6.14733104444041054861675057086, −5.42193682827957319356839285588, −4.74806516784472776963759442206, −3.70359520139580572439608105442, −1.84260010861586212484136754986, −1.02175075791668915720464370362,
1.02175075791668915720464370362, 1.84260010861586212484136754986, 3.70359520139580572439608105442, 4.74806516784472776963759442206, 5.42193682827957319356839285588, 6.14733104444041054861675057086, 7.27764101552667293667112897119, 8.177721846180402994333234296894, 8.946785705386854011519557198208, 9.465876405028484689637476517904
