| L(s) = 1 | + 0.829·2-s − 0.269·3-s − 1.31·4-s − 1.24·5-s − 0.223·6-s + 4.95·7-s − 2.74·8-s − 2.92·9-s − 1.03·10-s − 4.14·11-s + 0.353·12-s + 0.518·13-s + 4.10·14-s + 0.335·15-s + 0.348·16-s − 17-s − 2.42·18-s − 1.45·19-s + 1.63·20-s − 1.33·21-s − 3.43·22-s − 4.30·23-s + 0.740·24-s − 3.44·25-s + 0.429·26-s + 1.59·27-s − 6.50·28-s + ⋯ |
| L(s) = 1 | + 0.586·2-s − 0.155·3-s − 0.656·4-s − 0.556·5-s − 0.0912·6-s + 1.87·7-s − 0.970·8-s − 0.975·9-s − 0.326·10-s − 1.25·11-s + 0.102·12-s + 0.143·13-s + 1.09·14-s + 0.0866·15-s + 0.0871·16-s − 0.242·17-s − 0.572·18-s − 0.334·19-s + 0.365·20-s − 0.291·21-s − 0.733·22-s − 0.897·23-s + 0.151·24-s − 0.689·25-s + 0.0842·26-s + 0.307·27-s − 1.22·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 0.829T + 2T^{2} \) |
| 3 | \( 1 + 0.269T + 3T^{2} \) |
| 5 | \( 1 + 1.24T + 5T^{2} \) |
| 7 | \( 1 - 4.95T + 7T^{2} \) |
| 11 | \( 1 + 4.14T + 11T^{2} \) |
| 13 | \( 1 - 0.518T + 13T^{2} \) |
| 19 | \( 1 + 1.45T + 19T^{2} \) |
| 23 | \( 1 + 4.30T + 23T^{2} \) |
| 29 | \( 1 + 2.92T + 29T^{2} \) |
| 31 | \( 1 + 6.03T + 31T^{2} \) |
| 37 | \( 1 - 4.49T + 37T^{2} \) |
| 41 | \( 1 + 12.5T + 41T^{2} \) |
| 43 | \( 1 + 4.00T + 43T^{2} \) |
| 53 | \( 1 + 8.32T + 53T^{2} \) |
| 59 | \( 1 - 7.17T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 6.87T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 - 7.99T + 73T^{2} \) |
| 79 | \( 1 + 2.54T + 79T^{2} \) |
| 83 | \( 1 - 0.605T + 83T^{2} \) |
| 89 | \( 1 + 1.58T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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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
−9.940336499898374037877723668843, −8.559710603292493548094908678235, −8.313505565389215738521731719971, −7.50272183586563477303340325924, −5.88562742999163430846977708860, −5.22530229916776403755034872294, −4.55735525796986178724260282014, −3.51869776393656596291188623645, −2.10229431787684958591346764660, 0,
2.10229431787684958591346764660, 3.51869776393656596291188623645, 4.55735525796986178724260282014, 5.22530229916776403755034872294, 5.88562742999163430846977708860, 7.50272183586563477303340325924, 8.313505565389215738521731719971, 8.559710603292493548094908678235, 9.940336499898374037877723668843
