L(s) = 1 | + 4-s − 7-s + 13-s + 16-s + 2·19-s − 25-s − 28-s − 2·31-s − 2·43-s + 49-s + 52-s + 64-s − 2·73-s + 2·76-s − 2·79-s − 91-s + 2·97-s − 100-s − 112-s + ⋯ |
L(s) = 1 | + 4-s − 7-s + 13-s + 16-s + 2·19-s − 25-s − 28-s − 2·31-s − 2·43-s + 49-s + 52-s + 64-s − 2·73-s + 2·76-s − 2·79-s − 91-s + 2·97-s − 100-s − 112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.156262399\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.156262399\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 + T )^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 + T )^{2} \) |
| 79 | \( ( 1 + T )^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( ( 1 - T )^{2} \) |
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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
−10.38112261377548009711383915204, −9.734770557930096862349388829892, −8.821265993649184596618810903326, −7.64648652971990362637381209816, −7.03790083133763062798246867453, −6.07830468626135315869691362260, −5.44166535387810146752310137560, −3.68818235190388010834578147333, −3.07630926146048466467025302930, −1.59951812466743409116239825377,
1.59951812466743409116239825377, 3.07630926146048466467025302930, 3.68818235190388010834578147333, 5.44166535387810146752310137560, 6.07830468626135315869691362260, 7.03790083133763062798246867453, 7.64648652971990362637381209816, 8.821265993649184596618810903326, 9.734770557930096862349388829892, 10.38112261377548009711383915204