| L(s) = 1 | + 0.976·2-s − 6.99·3-s − 7.04·4-s − 1.89·5-s − 6.82·6-s + 11.4·7-s − 14.6·8-s + 21.9·9-s − 1.85·10-s + 24.2·11-s + 49.3·12-s − 51.2·13-s + 11.1·14-s + 13.2·15-s + 42.0·16-s + 83.4·17-s + 21.4·18-s + 150.·19-s + 13.3·20-s − 80.0·21-s + 23.6·22-s + 102.·24-s − 121.·25-s − 50.0·26-s + 35.3·27-s − 80.6·28-s − 252.·29-s + ⋯ |
| L(s) = 1 | + 0.345·2-s − 1.34·3-s − 0.880·4-s − 0.169·5-s − 0.464·6-s + 0.618·7-s − 0.649·8-s + 0.812·9-s − 0.0585·10-s + 0.664·11-s + 1.18·12-s − 1.09·13-s + 0.213·14-s + 0.228·15-s + 0.656·16-s + 1.19·17-s + 0.280·18-s + 1.82·19-s + 0.149·20-s − 0.832·21-s + 0.229·22-s + 0.873·24-s − 0.971·25-s − 0.377·26-s + 0.252·27-s − 0.544·28-s − 1.61·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 \) |
| good | 2 | \( 1 - 0.976T + 8T^{2} \) |
| 3 | \( 1 + 6.99T + 27T^{2} \) |
| 5 | \( 1 + 1.89T + 125T^{2} \) |
| 7 | \( 1 - 11.4T + 343T^{2} \) |
| 11 | \( 1 - 24.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 51.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 83.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 150.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 252.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 43.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 35.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 333.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 293.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 103.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 211.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 115.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 184.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 545.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 582.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 47.4T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.25e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 759.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 785.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 560.T + 9.12e5T^{2} \) |
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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
−9.874691830265283888977449560724, −9.463506324839619560546948480916, −8.030145903207590434768955111111, −7.24613393579190246171177339283, −5.84199384237263069962852602488, −5.33369681618928758344021549372, −4.53756335356326022132201825002, −3.37351880804525772638874757286, −1.24477364492803695997542771755, 0,
1.24477364492803695997542771755, 3.37351880804525772638874757286, 4.53756335356326022132201825002, 5.33369681618928758344021549372, 5.84199384237263069962852602488, 7.24613393579190246171177339283, 8.030145903207590434768955111111, 9.463506324839619560546948480916, 9.874691830265283888977449560724
