L(s) = 1 | − 1.19·2-s + 2.27·3-s − 0.574·4-s − 2.71·6-s + 2.19·7-s + 3.07·8-s + 2.16·9-s − 2.82·11-s − 1.30·12-s + 4.08·13-s − 2.62·14-s − 2.52·16-s − 0.382·17-s − 2.58·18-s + 4.99·21-s + 3.37·22-s + 2.97·23-s + 6.98·24-s − 4.87·26-s − 1.89·27-s − 1.26·28-s − 8.57·29-s + 4.69·31-s − 3.13·32-s − 6.42·33-s + 0.456·34-s − 1.24·36-s + ⋯ |
L(s) = 1 | − 0.844·2-s + 1.31·3-s − 0.287·4-s − 1.10·6-s + 0.830·7-s + 1.08·8-s + 0.722·9-s − 0.852·11-s − 0.376·12-s + 1.13·13-s − 0.700·14-s − 0.630·16-s − 0.0928·17-s − 0.609·18-s + 1.08·21-s + 0.719·22-s + 0.620·23-s + 1.42·24-s − 0.955·26-s − 0.364·27-s − 0.238·28-s − 1.59·29-s + 0.842·31-s − 0.554·32-s − 1.11·33-s + 0.0783·34-s − 0.207·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.058401308\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.058401308\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + 1.19T + 2T^{2} \) |
| 3 | \( 1 - 2.27T + 3T^{2} \) |
| 7 | \( 1 - 2.19T + 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 - 4.08T + 13T^{2} \) |
| 17 | \( 1 + 0.382T + 17T^{2} \) |
| 23 | \( 1 - 2.97T + 23T^{2} \) |
| 29 | \( 1 + 8.57T + 29T^{2} \) |
| 31 | \( 1 - 4.69T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 + 3.13T + 41T^{2} \) |
| 43 | \( 1 + 2.00T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 7.47T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 - 5.97T + 61T^{2} \) |
| 67 | \( 1 - 1.04T + 67T^{2} \) |
| 71 | \( 1 - 7.82T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 - 5.55T + 83T^{2} \) |
| 89 | \( 1 - 6.56T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−8.008969145001365683750461245006, −7.45619918011821325380668723168, −6.72274830172079890297056163611, −5.44199500678946979352807071322, −5.04157642126148041907309082809, −3.91423730635801127743564493906, −3.55611887689334075267309444570, −2.36473973259520842542601089614, −1.79334718723796537714382587646, −0.75950054406468337250790481063,
0.75950054406468337250790481063, 1.79334718723796537714382587646, 2.36473973259520842542601089614, 3.55611887689334075267309444570, 3.91423730635801127743564493906, 5.04157642126148041907309082809, 5.44199500678946979352807071322, 6.72274830172079890297056163611, 7.45619918011821325380668723168, 8.008969145001365683750461245006