L(s) = 1 | − 2.37·2-s − 0.696·3-s + 3.64·4-s − 2.62·5-s + 1.65·6-s − 3.90·8-s − 2.51·9-s + 6.23·10-s + 4.46·11-s − 2.53·12-s + 1.82·15-s + 1.98·16-s − 2.35·17-s + 5.97·18-s + 6.82·19-s − 9.56·20-s − 10.6·22-s + 1.91·23-s + 2.71·24-s + 1.89·25-s + 3.84·27-s + 7.24·29-s − 4.34·30-s + 7.55·31-s + 3.09·32-s − 3.11·33-s + 5.58·34-s + ⋯ |
L(s) = 1 | − 1.67·2-s − 0.402·3-s + 1.82·4-s − 1.17·5-s + 0.675·6-s − 1.37·8-s − 0.838·9-s + 1.97·10-s + 1.34·11-s − 0.732·12-s + 0.472·15-s + 0.495·16-s − 0.570·17-s + 1.40·18-s + 1.56·19-s − 2.13·20-s − 2.26·22-s + 0.398·23-s + 0.554·24-s + 0.378·25-s + 0.739·27-s + 1.34·29-s − 0.793·30-s + 1.35·31-s + 0.546·32-s − 0.541·33-s + 0.958·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5913676221\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5913676221\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 2.37T + 2T^{2} \) |
| 3 | \( 1 + 0.696T + 3T^{2} \) |
| 5 | \( 1 + 2.62T + 5T^{2} \) |
| 11 | \( 1 - 4.46T + 11T^{2} \) |
| 17 | \( 1 + 2.35T + 17T^{2} \) |
| 19 | \( 1 - 6.82T + 19T^{2} \) |
| 23 | \( 1 - 1.91T + 23T^{2} \) |
| 29 | \( 1 - 7.24T + 29T^{2} \) |
| 31 | \( 1 - 7.55T + 31T^{2} \) |
| 37 | \( 1 + 5.94T + 37T^{2} \) |
| 41 | \( 1 - 0.271T + 41T^{2} \) |
| 43 | \( 1 + 3.28T + 43T^{2} \) |
| 47 | \( 1 - 8.10T + 47T^{2} \) |
| 53 | \( 1 - 7.19T + 53T^{2} \) |
| 59 | \( 1 - 9.04T + 59T^{2} \) |
| 61 | \( 1 - 7.88T + 61T^{2} \) |
| 67 | \( 1 + 7.00T + 67T^{2} \) |
| 71 | \( 1 - 6.84T + 71T^{2} \) |
| 73 | \( 1 + 0.348T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 + 6.00T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−8.029936093745014560081295491894, −7.12834964971220382641184542481, −6.85144987954259450936235569062, −6.06443091711285152776515957931, −5.07452585396902599534226522114, −4.18654030973354813736719584105, −3.31116978702665780990491662573, −2.48759510977355697433417615062, −1.16922290492667178425994318248, −0.60219919581535590650681902505,
0.60219919581535590650681902505, 1.16922290492667178425994318248, 2.48759510977355697433417615062, 3.31116978702665780990491662573, 4.18654030973354813736719584105, 5.07452585396902599534226522114, 6.06443091711285152776515957931, 6.85144987954259450936235569062, 7.12834964971220382641184542481, 8.029936093745014560081295491894