L(s) = 1 | + 3-s + 9-s + 3·11-s + 13-s − 17-s + 19-s + 27-s − 5·29-s − 4·31-s + 3·33-s − 2·37-s + 39-s − 2·41-s + 10·43-s − 7·47-s − 51-s + 3·53-s + 57-s − 5·59-s − 61-s − 9·67-s + 11·71-s + 4·73-s − 10·79-s + 81-s + 8·83-s − 5·87-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.904·11-s + 0.277·13-s − 0.242·17-s + 0.229·19-s + 0.192·27-s − 0.928·29-s − 0.718·31-s + 0.522·33-s − 0.328·37-s + 0.160·39-s − 0.312·41-s + 1.52·43-s − 1.02·47-s − 0.140·51-s + 0.412·53-s + 0.132·57-s − 0.650·59-s − 0.128·61-s − 1.09·67-s + 1.30·71-s + 0.468·73-s − 1.12·79-s + 1/9·81-s + 0.878·83-s − 0.536·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 382200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 382200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 11 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−12.63207954014998, −12.39101991857392, −11.77638817841904, −11.21995334693124, −11.07539655689753, −10.38140488506447, −9.932864943208451, −9.418197979923490, −9.021544642895607, −8.799416447380324, −8.101210749985457, −7.735263455415706, −7.141131407381056, −6.877821837008071, −6.163915664662831, −5.867123860830336, −5.187603239466378, −4.687047132401934, −4.030567662869455, −3.752671464929354, −3.203224877881183, −2.619633341552724, −1.934139212341742, −1.533735974332764, −0.8566228010002268, 0,
0.8566228010002268, 1.533735974332764, 1.934139212341742, 2.619633341552724, 3.203224877881183, 3.752671464929354, 4.030567662869455, 4.687047132401934, 5.187603239466378, 5.867123860830336, 6.163915664662831, 6.877821837008071, 7.141131407381056, 7.735263455415706, 8.101210749985457, 8.799416447380324, 9.021544642895607, 9.418197979923490, 9.932864943208451, 10.38140488506447, 11.07539655689753, 11.21995334693124, 11.77638817841904, 12.39101991857392, 12.63207954014998