| L(s) = 1 | + 1.78·2-s − 1.55·3-s + 1.19·4-s − 2.61·5-s − 2.77·6-s − 1.43·8-s − 0.593·9-s − 4.66·10-s − 3.67·11-s − 1.85·12-s − 4.83·13-s + 4.04·15-s − 4.96·16-s − 2.98·17-s − 1.06·18-s − 4.34·19-s − 3.12·20-s − 6.56·22-s + 2.59·23-s + 2.22·24-s + 1.81·25-s − 8.65·26-s + 5.57·27-s − 1.91·29-s + 7.23·30-s − 1.05·31-s − 5.99·32-s + ⋯ |
| L(s) = 1 | + 1.26·2-s − 0.895·3-s + 0.598·4-s − 1.16·5-s − 1.13·6-s − 0.507·8-s − 0.197·9-s − 1.47·10-s − 1.10·11-s − 0.535·12-s − 1.34·13-s + 1.04·15-s − 1.24·16-s − 0.724·17-s − 0.250·18-s − 0.997·19-s − 0.698·20-s − 1.40·22-s + 0.541·23-s + 0.454·24-s + 0.362·25-s − 1.69·26-s + 1.07·27-s − 0.356·29-s + 1.32·30-s − 0.189·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02253081319\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02253081319\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 1.78T + 2T^{2} \) |
| 3 | \( 1 + 1.55T + 3T^{2} \) |
| 5 | \( 1 + 2.61T + 5T^{2} \) |
| 11 | \( 1 + 3.67T + 11T^{2} \) |
| 13 | \( 1 + 4.83T + 13T^{2} \) |
| 17 | \( 1 + 2.98T + 17T^{2} \) |
| 19 | \( 1 + 4.34T + 19T^{2} \) |
| 23 | \( 1 - 2.59T + 23T^{2} \) |
| 29 | \( 1 + 1.91T + 29T^{2} \) |
| 31 | \( 1 + 1.05T + 31T^{2} \) |
| 37 | \( 1 + 7.32T + 37T^{2} \) |
| 41 | \( 1 - 1.13T + 41T^{2} \) |
| 43 | \( 1 + 7.87T + 43T^{2} \) |
| 47 | \( 1 + 2.04T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 + 4.81T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 1.31T + 67T^{2} \) |
| 71 | \( 1 - 7.73T + 71T^{2} \) |
| 73 | \( 1 - 6.77T + 73T^{2} \) |
| 79 | \( 1 - 0.216T + 79T^{2} \) |
| 83 | \( 1 - 3.43T + 83T^{2} \) |
| 89 | \( 1 + 9.32T + 89T^{2} \) |
| 97 | \( 1 + 0.511T + 97T^{2} \) |
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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
−7.958661304899351627307153047846, −7.01352094723134159478391764654, −6.57950058624467540392191436000, −5.63632287325006880901598598529, −4.96710979895652962296234911898, −4.72392770022070562007361016822, −3.81269596316101762725884517627, −3.01054704867887093865472258456, −2.22401667516944744367819212276, −0.06163451785951328735651630834,
0.06163451785951328735651630834, 2.22401667516944744367819212276, 3.01054704867887093865472258456, 3.81269596316101762725884517627, 4.72392770022070562007361016822, 4.96710979895652962296234911898, 5.63632287325006880901598598529, 6.57950058624467540392191436000, 7.01352094723134159478391764654, 7.958661304899351627307153047846
