| L(s) = 1 | − 2-s + 4-s − 0.853·5-s − 7-s − 8-s + 0.853·10-s − 2.68·11-s + 4.68·13-s + 14-s + 16-s + 1.83·17-s − 4.97·19-s − 0.853·20-s + 2.68·22-s + 23-s − 4.27·25-s − 4.68·26-s − 28-s + 6.39·29-s + 3.83·31-s − 32-s − 1.83·34-s + 0.853·35-s + 5.66·37-s + 4.97·38-s + 0.853·40-s − 11.9·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.381·5-s − 0.377·7-s − 0.353·8-s + 0.269·10-s − 0.809·11-s + 1.29·13-s + 0.267·14-s + 0.250·16-s + 0.444·17-s − 1.14·19-s − 0.190·20-s + 0.572·22-s + 0.208·23-s − 0.854·25-s − 0.918·26-s − 0.188·28-s + 1.18·29-s + 0.688·31-s − 0.176·32-s − 0.314·34-s + 0.144·35-s + 0.931·37-s + 0.807·38-s + 0.134·40-s − 1.86·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 0.853T + 5T^{2} \) |
| 11 | \( 1 + 2.68T + 11T^{2} \) |
| 13 | \( 1 - 4.68T + 13T^{2} \) |
| 17 | \( 1 - 1.83T + 17T^{2} \) |
| 19 | \( 1 + 4.97T + 19T^{2} \) |
| 29 | \( 1 - 6.39T + 29T^{2} \) |
| 31 | \( 1 - 3.83T + 31T^{2} \) |
| 37 | \( 1 - 5.66T + 37T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 + 9.37T + 43T^{2} \) |
| 47 | \( 1 - 3.83T + 47T^{2} \) |
| 53 | \( 1 - 9.66T + 53T^{2} \) |
| 59 | \( 1 + 3.43T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 + 2.68T + 67T^{2} \) |
| 71 | \( 1 - 0.335T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 + 5.31T + 83T^{2} \) |
| 89 | \( 1 + 3.53T + 89T^{2} \) |
| 97 | \( 1 - 2.75T + 97T^{2} \) |
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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
−8.292605468995504320197189085482, −7.963851241773470232322160076206, −6.85753467825747300305787918999, −6.31030425620419160236368933146, −5.46202732442618015582551206122, −4.35314759777858323178540058410, −3.44219190766388119267394469761, −2.56261672569622925185910442403, −1.32678859191035213307827463277, 0,
1.32678859191035213307827463277, 2.56261672569622925185910442403, 3.44219190766388119267394469761, 4.35314759777858323178540058410, 5.46202732442618015582551206122, 6.31030425620419160236368933146, 6.85753467825747300305787918999, 7.963851241773470232322160076206, 8.292605468995504320197189085482
