L(s) = 1 | − 3-s − 7-s + 9-s + 11-s + 13-s + 5·17-s − 2·19-s + 21-s + 7·23-s − 27-s − 4·31-s − 33-s + 7·37-s − 39-s − 11·41-s + 6·43-s − 6·49-s − 5·51-s − 11·53-s + 2·57-s − 4·59-s − 7·61-s − 63-s + 8·67-s − 7·69-s + 9·71-s − 2·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s + 1.21·17-s − 0.458·19-s + 0.218·21-s + 1.45·23-s − 0.192·27-s − 0.718·31-s − 0.174·33-s + 1.15·37-s − 0.160·39-s − 1.71·41-s + 0.914·43-s − 6/7·49-s − 0.700·51-s − 1.51·53-s + 0.264·57-s − 0.520·59-s − 0.896·61-s − 0.125·63-s + 0.977·67-s − 0.842·69-s + 1.06·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−15.34158022165805, −14.84158054120077, −14.38572085042619, −13.70781954739892, −13.15914803312914, −12.54671959191572, −12.37542401274303, −11.53024525877928, −11.09879414847898, −10.66931512671427, −9.953362896673122, −9.473909050865506, −9.011680950722107, −8.182101621889797, −7.725564606918993, −6.968979106355290, −6.520659092207075, −5.951387643023977, −5.283043119389192, −4.800695465700473, −3.981980801431456, −3.352125981214564, −2.749565420913614, −1.640040952277126, −1.034194772828473, 0,
1.034194772828473, 1.640040952277126, 2.749565420913614, 3.352125981214564, 3.981980801431456, 4.800695465700473, 5.283043119389192, 5.951387643023977, 6.520659092207075, 6.968979106355290, 7.725564606918993, 8.182101621889797, 9.011680950722107, 9.473909050865506, 9.953362896673122, 10.66931512671427, 11.09879414847898, 11.53024525877928, 12.37542401274303, 12.54671959191572, 13.15914803312914, 13.70781954739892, 14.38572085042619, 14.84158054120077, 15.34158022165805