| L(s) = 1 | + 5·7-s + 11-s − 2·13-s + 3·17-s + 7·19-s + 6·23-s + 3·29-s + 7·31-s + 7·37-s − 6·41-s + 8·43-s − 6·47-s + 18·49-s − 3·53-s − 6·59-s − 61-s + 8·67-s + 3·71-s − 2·73-s + 5·77-s + 10·79-s + 6·83-s − 9·89-s − 10·91-s + 4·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 1.88·7-s + 0.301·11-s − 0.554·13-s + 0.727·17-s + 1.60·19-s + 1.25·23-s + 0.557·29-s + 1.25·31-s + 1.15·37-s − 0.937·41-s + 1.21·43-s − 0.875·47-s + 18/7·49-s − 0.412·53-s − 0.781·59-s − 0.128·61-s + 0.977·67-s + 0.356·71-s − 0.234·73-s + 0.569·77-s + 1.12·79-s + 0.658·83-s − 0.953·89-s − 1.04·91-s + 0.406·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.378289778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.378289778\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.62809238546506, −14.33765384256320, −13.92100850535240, −13.39407350412949, −12.59894523571868, −11.97711912762259, −11.76427402420299, −11.11775977422787, −10.81362317723511, −9.912995759934909, −9.634704170604087, −8.852342574173084, −8.342890152042899, −7.684105260857296, −7.540304120647040, −6.755427154377353, −5.993155390108844, −5.266017203031681, −4.872091927238811, −4.501011355175037, −3.536817784299096, −2.870076688776632, −2.154281670525517, −1.189292801594879, −0.9488909707861077,
0.9488909707861077, 1.189292801594879, 2.154281670525517, 2.870076688776632, 3.536817784299096, 4.501011355175037, 4.872091927238811, 5.266017203031681, 5.993155390108844, 6.755427154377353, 7.540304120647040, 7.684105260857296, 8.342890152042899, 8.852342574173084, 9.634704170604087, 9.912995759934909, 10.81362317723511, 11.11775977422787, 11.76427402420299, 11.97711912762259, 12.59894523571868, 13.39407350412949, 13.92100850535240, 14.33765384256320, 14.62809238546506
