| L(s) = 1 | + 2-s − 0.360·3-s + 4-s + 4.09·5-s − 0.360·6-s − 3.96·7-s + 8-s − 2.86·9-s + 4.09·10-s − 6.50·11-s − 0.360·12-s − 13-s − 3.96·14-s − 1.47·15-s + 16-s + 1.07·17-s − 2.86·18-s − 1.75·19-s + 4.09·20-s + 1.43·21-s − 6.50·22-s + 5.71·23-s − 0.360·24-s + 11.7·25-s − 26-s + 2.11·27-s − 3.96·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.208·3-s + 0.5·4-s + 1.82·5-s − 0.147·6-s − 1.49·7-s + 0.353·8-s − 0.956·9-s + 1.29·10-s − 1.96·11-s − 0.104·12-s − 0.277·13-s − 1.05·14-s − 0.381·15-s + 0.250·16-s + 0.261·17-s − 0.676·18-s − 0.402·19-s + 0.914·20-s + 0.312·21-s − 1.38·22-s + 1.19·23-s − 0.0736·24-s + 2.34·25-s − 0.196·26-s + 0.407·27-s − 0.749·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2678 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2678 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 3 | \( 1 + 0.360T + 3T^{2} \) |
| 5 | \( 1 - 4.09T + 5T^{2} \) |
| 7 | \( 1 + 3.96T + 7T^{2} \) |
| 11 | \( 1 + 6.50T + 11T^{2} \) |
| 17 | \( 1 - 1.07T + 17T^{2} \) |
| 19 | \( 1 + 1.75T + 19T^{2} \) |
| 23 | \( 1 - 5.71T + 23T^{2} \) |
| 29 | \( 1 + 5.58T + 29T^{2} \) |
| 31 | \( 1 + 8.57T + 31T^{2} \) |
| 37 | \( 1 - 4.73T + 37T^{2} \) |
| 41 | \( 1 + 5.00T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 + 0.493T + 47T^{2} \) |
| 53 | \( 1 - 2.16T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 + 6.29T + 61T^{2} \) |
| 67 | \( 1 - 3.45T + 67T^{2} \) |
| 71 | \( 1 + 3.56T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 + 6.57T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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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
−8.622779517013576062924270574522, −7.43193546037779003299188739923, −6.65642383774403088592281121976, −5.89621512098332811996719038558, −5.51025292635887501789351726398, −4.91976361826329759638256118216, −3.19201598386626682893451969174, −2.84644122407185267856831698901, −1.93791200479332381442051893488, 0,
1.93791200479332381442051893488, 2.84644122407185267856831698901, 3.19201598386626682893451969174, 4.91976361826329759638256118216, 5.51025292635887501789351726398, 5.89621512098332811996719038558, 6.65642383774403088592281121976, 7.43193546037779003299188739923, 8.622779517013576062924270574522
