| L(s) = 1 | − 1.55·2-s + 3-s + 0.427·4-s − 1.55·6-s − 3.87·7-s + 2.45·8-s + 9-s + 1.67·11-s + 0.427·12-s + 2.12·13-s + 6.03·14-s − 4.67·16-s + 5.31·17-s − 1.55·18-s − 8.33·19-s − 3.87·21-s − 2.60·22-s − 0.669·23-s + 2.45·24-s − 3.30·26-s + 27-s − 1.65·28-s + 3.15·29-s − 2.86·31-s + 2.37·32-s + 1.67·33-s − 8.28·34-s + ⋯ |
| L(s) = 1 | − 1.10·2-s + 0.577·3-s + 0.213·4-s − 0.636·6-s − 1.46·7-s + 0.866·8-s + 0.333·9-s + 0.504·11-s + 0.123·12-s + 0.588·13-s + 1.61·14-s − 1.16·16-s + 1.28·17-s − 0.367·18-s − 1.91·19-s − 0.845·21-s − 0.555·22-s − 0.139·23-s + 0.500·24-s − 0.648·26-s + 0.192·27-s − 0.312·28-s + 0.586·29-s − 0.514·31-s + 0.420·32-s + 0.291·33-s − 1.42·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 + 1.55T + 2T^{2} \) |
| 7 | \( 1 + 3.87T + 7T^{2} \) |
| 11 | \( 1 - 1.67T + 11T^{2} \) |
| 13 | \( 1 - 2.12T + 13T^{2} \) |
| 17 | \( 1 - 5.31T + 17T^{2} \) |
| 19 | \( 1 + 8.33T + 19T^{2} \) |
| 23 | \( 1 + 0.669T + 23T^{2} \) |
| 29 | \( 1 - 3.15T + 29T^{2} \) |
| 31 | \( 1 + 2.86T + 31T^{2} \) |
| 37 | \( 1 + 0.398T + 37T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 + 1.82T + 43T^{2} \) |
| 53 | \( 1 - 2.61T + 53T^{2} \) |
| 59 | \( 1 + 2.45T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 + 0.720T + 67T^{2} \) |
| 71 | \( 1 - 8.17T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 5.57T + 83T^{2} \) |
| 89 | \( 1 + 3.73T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−8.461546805522032591115686629783, −7.70730729723258214130909719836, −6.71453861266662171930827867658, −6.42007998836686693784334804255, −5.20009290756684480247679439491, −3.98718991319995705848367283765, −3.51503249851338020148234956465, −2.35364669433195254805938607527, −1.25842513740384926578519752777, 0,
1.25842513740384926578519752777, 2.35364669433195254805938607527, 3.51503249851338020148234956465, 3.98718991319995705848367283765, 5.20009290756684480247679439491, 6.42007998836686693784334804255, 6.71453861266662171930827867658, 7.70730729723258214130909719836, 8.461546805522032591115686629783
