| L(s) = 1 | + 1.82·2-s − 1.94·3-s + 1.33·4-s − 2.55·5-s − 3.55·6-s − 2.25·7-s − 1.21·8-s + 0.800·9-s − 4.65·10-s − 11-s − 2.60·12-s − 5.29·13-s − 4.12·14-s + 4.97·15-s − 4.88·16-s − 0.162·17-s + 1.46·18-s + 5.27·19-s − 3.40·20-s + 4.40·21-s − 1.82·22-s + 1.31·23-s + 2.36·24-s + 1.50·25-s − 9.66·26-s + 4.28·27-s − 3.01·28-s + ⋯ |
| L(s) = 1 | + 1.29·2-s − 1.12·3-s + 0.667·4-s − 1.14·5-s − 1.45·6-s − 0.854·7-s − 0.429·8-s + 0.266·9-s − 1.47·10-s − 0.301·11-s − 0.751·12-s − 1.46·13-s − 1.10·14-s + 1.28·15-s − 1.22·16-s − 0.0393·17-s + 0.344·18-s + 1.21·19-s − 0.761·20-s + 0.961·21-s − 0.389·22-s + 0.273·23-s + 0.483·24-s + 0.301·25-s − 1.89·26-s + 0.825·27-s − 0.569·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2563 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2563 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6826752837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6826752837\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 233 | \( 1 - T \) |
| good | 2 | \( 1 - 1.82T + 2T^{2} \) |
| 3 | \( 1 + 1.94T + 3T^{2} \) |
| 5 | \( 1 + 2.55T + 5T^{2} \) |
| 7 | \( 1 + 2.25T + 7T^{2} \) |
| 13 | \( 1 + 5.29T + 13T^{2} \) |
| 17 | \( 1 + 0.162T + 17T^{2} \) |
| 19 | \( 1 - 5.27T + 19T^{2} \) |
| 23 | \( 1 - 1.31T + 23T^{2} \) |
| 29 | \( 1 + 6.69T + 29T^{2} \) |
| 31 | \( 1 - 3.95T + 31T^{2} \) |
| 37 | \( 1 - 4.25T + 37T^{2} \) |
| 41 | \( 1 + 5.70T + 41T^{2} \) |
| 43 | \( 1 - 7.30T + 43T^{2} \) |
| 47 | \( 1 - 3.66T + 47T^{2} \) |
| 53 | \( 1 - 8.78T + 53T^{2} \) |
| 59 | \( 1 + 3.99T + 59T^{2} \) |
| 61 | \( 1 + 5.14T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 5.89T + 73T^{2} \) |
| 79 | \( 1 + 8.33T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 + 0.443T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−9.003971963577314541241179039150, −7.70829839727305237209763621025, −7.19156832701410079569800966066, −6.32885160388743885361210979310, −5.58973296748623772781815359246, −4.98210175121108719208738443674, −4.28388741708035559296453887105, −3.37474553284390840955778798939, −2.65228007648775633895035446990, −0.42500900683030619202555250330,
0.42500900683030619202555250330, 2.65228007648775633895035446990, 3.37474553284390840955778798939, 4.28388741708035559296453887105, 4.98210175121108719208738443674, 5.58973296748623772781815359246, 6.32885160388743885361210979310, 7.19156832701410079569800966066, 7.70829839727305237209763621025, 9.003971963577314541241179039150
