L(s) = 1 | + 0.0509·2-s + 3-s − 1.99·4-s + 3.95·5-s + 0.0509·6-s + 1.89·7-s − 0.203·8-s + 9-s + 0.201·10-s + 5.26·11-s − 1.99·12-s − 2.15·13-s + 0.0963·14-s + 3.95·15-s + 3.98·16-s + 17-s + 0.0509·18-s + 3.28·19-s − 7.90·20-s + 1.89·21-s + 0.268·22-s − 2.60·23-s − 0.203·24-s + 10.6·25-s − 0.109·26-s + 27-s − 3.78·28-s + ⋯ |
L(s) = 1 | + 0.0360·2-s + 0.577·3-s − 0.998·4-s + 1.77·5-s + 0.0207·6-s + 0.715·7-s − 0.0719·8-s + 0.333·9-s + 0.0637·10-s + 1.58·11-s − 0.576·12-s − 0.597·13-s + 0.0257·14-s + 1.02·15-s + 0.996·16-s + 0.242·17-s + 0.0120·18-s + 0.753·19-s − 1.76·20-s + 0.413·21-s + 0.0571·22-s − 0.542·23-s − 0.0415·24-s + 2.13·25-s − 0.0215·26-s + 0.192·27-s − 0.714·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.410683098\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.410683098\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
good | 2 | \( 1 - 0.0509T + 2T^{2} \) |
| 5 | \( 1 - 3.95T + 5T^{2} \) |
| 7 | \( 1 - 1.89T + 7T^{2} \) |
| 11 | \( 1 - 5.26T + 11T^{2} \) |
| 13 | \( 1 + 2.15T + 13T^{2} \) |
| 19 | \( 1 - 3.28T + 19T^{2} \) |
| 23 | \( 1 + 2.60T + 23T^{2} \) |
| 29 | \( 1 - 1.79T + 29T^{2} \) |
| 31 | \( 1 - 1.06T + 31T^{2} \) |
| 37 | \( 1 + 6.27T + 37T^{2} \) |
| 41 | \( 1 - 8.76T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 - 6.79T + 47T^{2} \) |
| 53 | \( 1 + 6.84T + 53T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 + 0.635T + 71T^{2} \) |
| 73 | \( 1 + 3.45T + 73T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 + 6.91T + 89T^{2} \) |
| 97 | \( 1 - 2.23T + 97T^{2} \) |
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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
−8.693213940711212893080143703932, −7.88587938832406321555068793832, −6.91644975958720740256722511401, −6.14973382376852510127178553228, −5.37286612157629890961086029520, −4.75972067763705045183844944856, −3.90383521402293004055143394381, −2.91257986476183056780436559514, −1.79394432209538092529775278829, −1.19070889481419289845903346318,
1.19070889481419289845903346318, 1.79394432209538092529775278829, 2.91257986476183056780436559514, 3.90383521402293004055143394381, 4.75972067763705045183844944856, 5.37286612157629890961086029520, 6.14973382376852510127178553228, 6.91644975958720740256722511401, 7.88587938832406321555068793832, 8.693213940711212893080143703932