L(s) = 1 | + 7-s − 11-s − 2·13-s − 6·17-s − 8·19-s + 6·23-s − 6·29-s − 2·31-s − 2·37-s + 8·43-s + 12·47-s + 49-s + 6·53-s + 6·59-s + 8·61-s + 2·67-s + 10·73-s − 77-s − 8·79-s − 12·83-s − 6·89-s − 2·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
L(s) = 1 | + 0.377·7-s − 0.301·11-s − 0.554·13-s − 1.45·17-s − 1.83·19-s + 1.25·23-s − 1.11·29-s − 0.359·31-s − 0.328·37-s + 1.21·43-s + 1.75·47-s + 1/7·49-s + 0.824·53-s + 0.781·59-s + 1.02·61-s + 0.244·67-s + 1.17·73-s − 0.113·77-s − 0.900·79-s − 1.31·83-s − 0.635·89-s − 0.209·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 277200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.198329606\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.198329606\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.71516607183163, −12.47868988111706, −11.82394741577194, −11.16487319581930, −10.93139060300361, −10.70078097576279, −10.01110812399587, −9.471604638323433, −8.967194551275216, −8.585830702050257, −8.296420843924237, −7.476338390748605, −7.039731494745515, −6.877531475070395, −6.060253472081995, −5.632447971357574, −5.115955218001409, −4.518315317519934, −4.131743103187260, −3.697168628382965, −2.721822836754500, −2.330310536864193, −1.995153697805568, −1.086339091779204, −0.3068093885633660,
0.3068093885633660, 1.086339091779204, 1.995153697805568, 2.330310536864193, 2.721822836754500, 3.697168628382965, 4.131743103187260, 4.518315317519934, 5.115955218001409, 5.632447971357574, 6.060253472081995, 6.877531475070395, 7.039731494745515, 7.476338390748605, 8.296420843924237, 8.585830702050257, 8.967194551275216, 9.471604638323433, 10.01110812399587, 10.70078097576279, 10.93139060300361, 11.16487319581930, 11.82394741577194, 12.47868988111706, 12.71516607183163