| L(s) = 1 | + 2.12·2-s + 2.71·3-s + 2.52·4-s − 1.05·5-s + 5.77·6-s + 4.69·7-s + 1.11·8-s + 4.35·9-s − 2.24·10-s − 3.29·11-s + 6.85·12-s − 5.93·13-s + 9.97·14-s − 2.85·15-s − 2.67·16-s − 17-s + 9.27·18-s − 7.17·19-s − 2.66·20-s + 12.7·21-s − 7.00·22-s + 4.65·23-s + 3.03·24-s − 3.88·25-s − 12.6·26-s + 3.68·27-s + 11.8·28-s + ⋯ |
| L(s) = 1 | + 1.50·2-s + 1.56·3-s + 1.26·4-s − 0.471·5-s + 2.35·6-s + 1.77·7-s + 0.395·8-s + 1.45·9-s − 0.709·10-s − 0.992·11-s + 1.97·12-s − 1.64·13-s + 2.66·14-s − 0.738·15-s − 0.668·16-s − 0.242·17-s + 2.18·18-s − 1.64·19-s − 0.595·20-s + 2.77·21-s − 1.49·22-s + 0.970·23-s + 0.619·24-s − 0.777·25-s − 2.47·26-s + 0.708·27-s + 2.23·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.219494257\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.219494257\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - 2.12T + 2T^{2} \) |
| 3 | \( 1 - 2.71T + 3T^{2} \) |
| 5 | \( 1 + 1.05T + 5T^{2} \) |
| 7 | \( 1 - 4.69T + 7T^{2} \) |
| 11 | \( 1 + 3.29T + 11T^{2} \) |
| 13 | \( 1 + 5.93T + 13T^{2} \) |
| 19 | \( 1 + 7.17T + 19T^{2} \) |
| 23 | \( 1 - 4.65T + 23T^{2} \) |
| 29 | \( 1 + 0.655T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 + 3.32T + 37T^{2} \) |
| 41 | \( 1 - 8.61T + 41T^{2} \) |
| 43 | \( 1 - 6.45T + 43T^{2} \) |
| 53 | \( 1 - 2.57T + 53T^{2} \) |
| 59 | \( 1 - 4.85T + 59T^{2} \) |
| 61 | \( 1 - 2.83T + 61T^{2} \) |
| 67 | \( 1 - 1.69T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 1.43T + 73T^{2} \) |
| 79 | \( 1 + 3.72T + 79T^{2} \) |
| 83 | \( 1 - 6.57T + 83T^{2} \) |
| 89 | \( 1 - 6.81T + 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
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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
−10.46401300957414935720772657217, −9.224286624430047069828985495090, −8.245802563809250144198713517734, −7.81978196837007301118928501067, −6.93922956490329764593434354238, −5.36815408996788893804941300182, −4.58246460672211950265983764428, −4.07486800883964427625901210661, −2.58573303350657552703306852109, −2.28301408139589334902095997024,
2.28301408139589334902095997024, 2.58573303350657552703306852109, 4.07486800883964427625901210661, 4.58246460672211950265983764428, 5.36815408996788893804941300182, 6.93922956490329764593434354238, 7.81978196837007301118928501067, 8.245802563809250144198713517734, 9.224286624430047069828985495090, 10.46401300957414935720772657217
