| L(s) = 1 | + 5.53·2-s − 9.29·3-s + 22.6·4-s + 3.61·5-s − 51.4·6-s + 7·7-s + 81.1·8-s + 59.3·9-s + 20.0·10-s + 0.694·11-s − 210.·12-s + 20.9·13-s + 38.7·14-s − 33.6·15-s + 267.·16-s − 46.8·17-s + 328.·18-s + 154.·19-s + 81.9·20-s − 65.0·21-s + 3.84·22-s + 23·23-s − 754.·24-s − 111.·25-s + 115.·26-s − 300.·27-s + 158.·28-s + ⋯ |
| L(s) = 1 | + 1.95·2-s − 1.78·3-s + 2.83·4-s + 0.323·5-s − 3.50·6-s + 0.377·7-s + 3.58·8-s + 2.19·9-s + 0.633·10-s + 0.0190·11-s − 5.06·12-s + 0.446·13-s + 0.739·14-s − 0.578·15-s + 4.18·16-s − 0.667·17-s + 4.30·18-s + 1.86·19-s + 0.916·20-s − 0.676·21-s + 0.0372·22-s + 0.208·23-s − 6.41·24-s − 0.895·25-s + 0.873·26-s − 2.14·27-s + 1.07·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.724246060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.724246060\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 - 5.53T + 8T^{2} \) |
| 3 | \( 1 + 9.29T + 27T^{2} \) |
| 5 | \( 1 - 3.61T + 125T^{2} \) |
| 11 | \( 1 - 0.694T + 1.33e3T^{2} \) |
| 13 | \( 1 - 20.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 46.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 154.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 211.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 38.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 378.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 419.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 49.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 63.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 207.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 145.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 282.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 265.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 848.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 19.7T + 3.89e5T^{2} \) |
| 79 | \( 1 + 115.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.30e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 318.T + 9.12e5T^{2} \) |
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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
−12.34532414017893698303308188993, −11.57784943029116616944269670348, −11.12072377355390519876702416816, −10.06611974741486311230476921464, −7.40189394942123686107720737220, −6.52834230114434864280679196079, −5.52609286939167776342260461328, −5.04908363739325738464022464764, −3.73306016423828474815245248575, −1.61488667653108141221149060681,
1.61488667653108141221149060681, 3.73306016423828474815245248575, 5.04908363739325738464022464764, 5.52609286939167776342260461328, 6.52834230114434864280679196079, 7.40189394942123686107720737220, 10.06611974741486311230476921464, 11.12072377355390519876702416816, 11.57784943029116616944269670348, 12.34532414017893698303308188993
