L(s) = 1 | − 3-s + 9-s − 11-s − 3·13-s + 4·17-s − 19-s − 2·23-s − 27-s − 29-s − 4·31-s + 33-s + 3·37-s + 3·39-s − 4·41-s − 2·43-s + 5·47-s − 4·51-s − 2·53-s + 57-s − 9·59-s + 2·61-s + 3·67-s + 2·69-s + 12·71-s + 7·73-s − 8·79-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.301·11-s − 0.832·13-s + 0.970·17-s − 0.229·19-s − 0.417·23-s − 0.192·27-s − 0.185·29-s − 0.718·31-s + 0.174·33-s + 0.493·37-s + 0.480·39-s − 0.624·41-s − 0.304·43-s + 0.729·47-s − 0.560·51-s − 0.274·53-s + 0.132·57-s − 1.17·59-s + 0.256·61-s + 0.366·67-s + 0.240·69-s + 1.42·71-s + 0.819·73-s − 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.90757332191871, −12.27623955902151, −12.01616731062932, −11.53260459120277, −11.03417319139294, −10.47247362340614, −10.28561149462192, −9.562777729781201, −9.430459956962298, −8.733425311095331, −8.108289805703019, −7.679313852737722, −7.413977950973864, −6.687927375722754, −6.357809613740229, −5.754009044959167, −5.195065485436791, −5.042416643833795, −4.289618833569675, −3.784470115687064, −3.253290473424743, −2.566825408769767, −2.041347515268345, −1.398186702256106, −0.6591492494238998, 0,
0.6591492494238998, 1.398186702256106, 2.041347515268345, 2.566825408769767, 3.253290473424743, 3.784470115687064, 4.289618833569675, 5.042416643833795, 5.195065485436791, 5.754009044959167, 6.357809613740229, 6.687927375722754, 7.413977950973864, 7.679313852737722, 8.108289805703019, 8.733425311095331, 9.430459956962298, 9.562777729781201, 10.28561149462192, 10.47247362340614, 11.03417319139294, 11.53260459120277, 12.01616731062932, 12.27623955902151, 12.90757332191871