L(s) = 1 | + 2.30·2-s + 1.58·3-s + 3.30·4-s − 1.03·5-s + 3.65·6-s + 2.92·7-s + 3.00·8-s − 0.481·9-s − 2.39·10-s − 11-s + 5.24·12-s + 4.96·13-s + 6.73·14-s − 1.64·15-s + 0.308·16-s − 0.923·17-s − 1.10·18-s + 2.42·19-s − 3.43·20-s + 4.64·21-s − 2.30·22-s + 7.67·23-s + 4.76·24-s − 3.92·25-s + 11.4·26-s − 5.52·27-s + 9.66·28-s + ⋯ |
L(s) = 1 | + 1.62·2-s + 0.916·3-s + 1.65·4-s − 0.464·5-s + 1.49·6-s + 1.10·7-s + 1.06·8-s − 0.160·9-s − 0.756·10-s − 0.301·11-s + 1.51·12-s + 1.37·13-s + 1.80·14-s − 0.425·15-s + 0.0772·16-s − 0.223·17-s − 0.261·18-s + 0.556·19-s − 0.767·20-s + 1.01·21-s − 0.491·22-s + 1.60·23-s + 0.972·24-s − 0.784·25-s + 2.24·26-s − 1.06·27-s + 1.82·28-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
\(L(1)\) |
\(\approx\) |
\(5.739220904\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.739220904\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.30T + 2T^{2} \) |
| 3 | \( 1 - 1.58T + 3T^{2} \) |
| 5 | \( 1 + 1.03T + 5T^{2} \) |
| 7 | \( 1 - 2.92T + 7T^{2} \) |
| 13 | \( 1 - 4.96T + 13T^{2} \) |
| 17 | \( 1 + 0.923T + 17T^{2} \) |
| 19 | \( 1 - 2.42T + 19T^{2} \) |
| 23 | \( 1 - 7.67T + 23T^{2} \) |
| 29 | \( 1 - 0.972T + 29T^{2} \) |
| 31 | \( 1 + 7.30T + 31T^{2} \) |
| 37 | \( 1 + 5.68T + 37T^{2} \) |
| 41 | \( 1 - 7.94T + 41T^{2} \) |
| 43 | \( 1 - 7.89T + 43T^{2} \) |
| 47 | \( 1 + 6.51T + 47T^{2} \) |
| 53 | \( 1 + 6.14T + 53T^{2} \) |
| 59 | \( 1 - 4.98T + 59T^{2} \) |
| 61 | \( 1 - 7.60T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 0.558T + 71T^{2} \) |
| 73 | \( 1 + 9.66T + 73T^{2} \) |
| 79 | \( 1 + 5.03T + 79T^{2} \) |
| 83 | \( 1 - 6.86T + 83T^{2} \) |
| 89 | \( 1 - 0.308T + 89T^{2} \) |
| 97 | \( 1 + 6.16T + 97T^{2} \) |
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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
−9.262472296524611205887440345204, −8.612220578217796485228532883988, −7.81196866989769748289960177955, −7.09508367172804464199630004083, −5.90228082796799574207285365912, −5.25644005568198635050866213191, −4.30438892067286142020471296204, −3.56337095633470232823965749599, −2.82268731749055668678857498951, −1.66322937452274652005312330577,
1.66322937452274652005312330577, 2.82268731749055668678857498951, 3.56337095633470232823965749599, 4.30438892067286142020471296204, 5.25644005568198635050866213191, 5.90228082796799574207285365912, 7.09508367172804464199630004083, 7.81196866989769748289960177955, 8.612220578217796485228532883988, 9.262472296524611205887440345204