| L(s) = 1 | − 1.13·2-s + 1.24·3-s − 0.722·4-s − 3.15·5-s − 1.40·6-s − 4.57·7-s + 3.07·8-s − 1.45·9-s + 3.57·10-s − 3.66·11-s − 0.898·12-s − 13-s + 5.17·14-s − 3.92·15-s − 2.03·16-s − 4.09·17-s + 1.64·18-s − 0.591·19-s + 2.28·20-s − 5.69·21-s + 4.13·22-s + 6.91·23-s + 3.82·24-s + 4.97·25-s + 1.13·26-s − 5.53·27-s + 3.30·28-s + ⋯ |
| L(s) = 1 | − 0.799·2-s + 0.718·3-s − 0.361·4-s − 1.41·5-s − 0.574·6-s − 1.72·7-s + 1.08·8-s − 0.484·9-s + 1.12·10-s − 1.10·11-s − 0.259·12-s − 0.277·13-s + 1.38·14-s − 1.01·15-s − 0.508·16-s − 0.993·17-s + 0.386·18-s − 0.135·19-s + 0.510·20-s − 1.24·21-s + 0.882·22-s + 1.44·23-s + 0.781·24-s + 0.994·25-s + 0.221·26-s − 1.06·27-s + 0.624·28-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
| \(L(1)\) |
\(\approx\) |
\(0.2353328010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2353328010\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.13T + 2T^{2} \) |
| 3 | \( 1 - 1.24T + 3T^{2} \) |
| 5 | \( 1 + 3.15T + 5T^{2} \) |
| 7 | \( 1 + 4.57T + 7T^{2} \) |
| 11 | \( 1 + 3.66T + 11T^{2} \) |
| 17 | \( 1 + 4.09T + 17T^{2} \) |
| 19 | \( 1 + 0.591T + 19T^{2} \) |
| 23 | \( 1 - 6.91T + 23T^{2} \) |
| 29 | \( 1 - 1.63T + 29T^{2} \) |
| 31 | \( 1 - 3.68T + 31T^{2} \) |
| 37 | \( 1 + 6.63T + 37T^{2} \) |
| 41 | \( 1 + 6.90T + 41T^{2} \) |
| 43 | \( 1 - 4.18T + 43T^{2} \) |
| 47 | \( 1 - 0.196T + 47T^{2} \) |
| 53 | \( 1 - 3.72T + 53T^{2} \) |
| 59 | \( 1 - 9.88T + 59T^{2} \) |
| 61 | \( 1 + 6.09T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 4.91T + 71T^{2} \) |
| 73 | \( 1 - 1.46T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 - 8.83T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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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
−9.420491144862672707801329258549, −8.685763004129163078934795273591, −8.300500529200881334044361395325, −7.34396114885871202765073090393, −6.80042452981398596700240069597, −5.33662141580053219136116942621, −4.26033310552593640958202324882, −3.35965294260439942794877146729, −2.62920119788896114099363174540, −0.35926842430367942570750760550,
0.35926842430367942570750760550, 2.62920119788896114099363174540, 3.35965294260439942794877146729, 4.26033310552593640958202324882, 5.33662141580053219136116942621, 6.80042452981398596700240069597, 7.34396114885871202765073090393, 8.300500529200881334044361395325, 8.685763004129163078934795273591, 9.420491144862672707801329258549
