| L(s) = 1 | − 91·7-s + 174·11-s + 785·13-s − 1.79e3·17-s − 925·19-s + 2.34e3·23-s + 726·29-s − 811·31-s + 7.92e3·37-s + 360·41-s − 4.95e3·43-s − 9.90e3·47-s − 8.52e3·49-s + 8.29e3·53-s − 7.01e3·59-s − 5.14e4·61-s + 581·67-s + 5.65e4·71-s − 4.24e4·73-s − 1.58e4·77-s − 2.89e4·79-s + 1.04e5·83-s + 1.18e5·89-s − 7.14e4·91-s + 1.10e5·97-s + 1.93e5·101-s − 3.76e4·103-s + ⋯ |
| L(s) = 1 | − 0.701·7-s + 0.433·11-s + 1.28·13-s − 1.50·17-s − 0.587·19-s + 0.924·23-s + 0.160·29-s − 0.151·31-s + 0.951·37-s + 0.0334·41-s − 0.408·43-s − 0.654·47-s − 0.507·49-s + 0.405·53-s − 0.262·59-s − 1.76·61-s + 0.0158·67-s + 1.33·71-s − 0.932·73-s − 0.304·77-s − 0.521·79-s + 1.66·83-s + 1.58·89-s − 0.904·91-s + 1.18·97-s + 1.88·101-s − 0.349·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.835859668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.835859668\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 13 p T + p^{5} T^{2} \) |
| 11 | \( 1 - 174 T + p^{5} T^{2} \) |
| 13 | \( 1 - 785 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1794 T + p^{5} T^{2} \) |
| 19 | \( 1 + 925 T + p^{5} T^{2} \) |
| 23 | \( 1 - 102 p T + p^{5} T^{2} \) |
| 29 | \( 1 - 726 T + p^{5} T^{2} \) |
| 31 | \( 1 + 811 T + p^{5} T^{2} \) |
| 37 | \( 1 - 7922 T + p^{5} T^{2} \) |
| 41 | \( 1 - 360 T + p^{5} T^{2} \) |
| 43 | \( 1 + 4951 T + p^{5} T^{2} \) |
| 47 | \( 1 + 9906 T + p^{5} T^{2} \) |
| 53 | \( 1 - 8292 T + p^{5} T^{2} \) |
| 59 | \( 1 + 7014 T + p^{5} T^{2} \) |
| 61 | \( 1 + 51433 T + p^{5} T^{2} \) |
| 67 | \( 1 - 581 T + p^{5} T^{2} \) |
| 71 | \( 1 - 56520 T + p^{5} T^{2} \) |
| 73 | \( 1 + 42478 T + p^{5} T^{2} \) |
| 79 | \( 1 + 28912 T + p^{5} T^{2} \) |
| 83 | \( 1 - 104586 T + p^{5} T^{2} \) |
| 89 | \( 1 - 118080 T + p^{5} T^{2} \) |
| 97 | \( 1 - 110273 T + p^{5} T^{2} \) |
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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
−9.113923945252563766744140735200, −8.811859703473545820210787585378, −7.68117895437131826682200114512, −6.41685595692483687666022074327, −6.35697815948279851313609936141, −4.88182631219498919476946411764, −3.95329413557830775895178098738, −3.03572250883302896260112419981, −1.81892525918228175346908635741, −0.60235921098571195360278357403,
0.60235921098571195360278357403, 1.81892525918228175346908635741, 3.03572250883302896260112419981, 3.95329413557830775895178098738, 4.88182631219498919476946411764, 6.35697815948279851313609936141, 6.41685595692483687666022074327, 7.68117895437131826682200114512, 8.811859703473545820210787585378, 9.113923945252563766744140735200
