L(s) = 1 | + 1.54·2-s + 2.73·3-s + 0.387·4-s − 3.25·5-s + 4.22·6-s − 3.29·7-s − 2.49·8-s + 4.47·9-s − 5.02·10-s − 11-s + 1.05·12-s − 5.51·13-s − 5.09·14-s − 8.89·15-s − 4.62·16-s + 0.673·17-s + 6.91·18-s + 3.11·19-s − 1.25·20-s − 9.00·21-s − 1.54·22-s + 0.320·23-s − 6.81·24-s + 5.58·25-s − 8.51·26-s + 4.02·27-s − 1.27·28-s + ⋯ |
L(s) = 1 | + 1.09·2-s + 1.57·3-s + 0.193·4-s − 1.45·5-s + 1.72·6-s − 1.24·7-s − 0.881·8-s + 1.49·9-s − 1.58·10-s − 0.301·11-s + 0.305·12-s − 1.52·13-s − 1.36·14-s − 2.29·15-s − 1.15·16-s + 0.163·17-s + 1.62·18-s + 0.713·19-s − 0.281·20-s − 1.96·21-s − 0.329·22-s + 0.0669·23-s − 1.39·24-s + 1.11·25-s − 1.67·26-s + 0.774·27-s − 0.241·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 131 | \( 1 + T \) |
good | 2 | \( 1 - 1.54T + 2T^{2} \) |
| 3 | \( 1 - 2.73T + 3T^{2} \) |
| 5 | \( 1 + 3.25T + 5T^{2} \) |
| 7 | \( 1 + 3.29T + 7T^{2} \) |
| 13 | \( 1 + 5.51T + 13T^{2} \) |
| 17 | \( 1 - 0.673T + 17T^{2} \) |
| 19 | \( 1 - 3.11T + 19T^{2} \) |
| 23 | \( 1 - 0.320T + 23T^{2} \) |
| 29 | \( 1 + 1.44T + 29T^{2} \) |
| 31 | \( 1 + 1.86T + 31T^{2} \) |
| 37 | \( 1 + 5.81T + 37T^{2} \) |
| 41 | \( 1 + 2.70T + 41T^{2} \) |
| 43 | \( 1 - 7.48T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 1.04T + 53T^{2} \) |
| 59 | \( 1 + 5.63T + 59T^{2} \) |
| 61 | \( 1 - 4.55T + 61T^{2} \) |
| 67 | \( 1 + 15.7T + 67T^{2} \) |
| 71 | \( 1 + 7.75T + 71T^{2} \) |
| 73 | \( 1 - 2.58T + 73T^{2} \) |
| 79 | \( 1 - 16.3T + 79T^{2} \) |
| 83 | \( 1 - 4.48T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 + 18.8T + 97T^{2} \) |
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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
−9.218289170126169588090260073179, −8.244362459046360220674573913487, −7.44341184057240131459873630520, −6.97571647473788215698804883579, −5.58100274313053210773588208415, −4.52008925368358711323425562433, −3.78980778730348711137413069238, −3.16591450829839614618014559042, −2.57522998017823687331072637192, 0,
2.57522998017823687331072637192, 3.16591450829839614618014559042, 3.78980778730348711137413069238, 4.52008925368358711323425562433, 5.58100274313053210773588208415, 6.97571647473788215698804883579, 7.44341184057240131459873630520, 8.244362459046360220674573913487, 9.218289170126169588090260073179