L(s) = 1 | − 2-s + 4-s + 3·5-s − 4·7-s − 8-s − 3·10-s − 4·13-s + 4·14-s + 16-s + 6·17-s + 5·19-s + 3·20-s + 4·25-s + 4·26-s − 4·28-s − 3·29-s − 4·31-s − 32-s − 6·34-s − 12·35-s − 10·37-s − 5·38-s − 3·40-s − 12·41-s − 43-s + 12·47-s + 9·49-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s − 1.51·7-s − 0.353·8-s − 0.948·10-s − 1.10·13-s + 1.06·14-s + 1/4·16-s + 1.45·17-s + 1.14·19-s + 0.670·20-s + 4/5·25-s + 0.784·26-s − 0.755·28-s − 0.557·29-s − 0.718·31-s − 0.176·32-s − 1.02·34-s − 2.02·35-s − 1.64·37-s − 0.811·38-s − 0.474·40-s − 1.87·41-s − 0.152·43-s + 1.75·47-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−7.33906114958670700822803097319, −7.18155889861700440952376595929, −6.28961395379311355991730150496, −5.54142153207581153802829036406, −5.25610302207423719401686501704, −3.64260116382931221040510961921, −3.05097655607076376010780973398, −2.23052841750958699618848771198, −1.28303245489691767449128898788, 0,
1.28303245489691767449128898788, 2.23052841750958699618848771198, 3.05097655607076376010780973398, 3.64260116382931221040510961921, 5.25610302207423719401686501704, 5.54142153207581153802829036406, 6.28961395379311355991730150496, 7.18155889861700440952376595929, 7.33906114958670700822803097319