| L(s) = 1 | + 11-s − 2·13-s − 2·17-s − 8·19-s − 4·23-s − 2·29-s − 8·31-s + 2·37-s − 6·41-s + 8·43-s + 4·47-s − 7·49-s + 2·53-s + 4·59-s − 6·61-s − 12·67-s − 12·71-s − 2·73-s − 4·83-s + 6·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.301·11-s − 0.554·13-s − 0.485·17-s − 1.83·19-s − 0.834·23-s − 0.371·29-s − 1.43·31-s + 0.328·37-s − 0.937·41-s + 1.21·43-s + 0.583·47-s − 49-s + 0.274·53-s + 0.520·59-s − 0.768·61-s − 1.46·67-s − 1.42·71-s − 0.234·73-s − 0.439·83-s + 0.635·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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
| \(L(1)\) |
\(\approx\) |
\(0.8070224999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8070224999\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−14.77842230353687, −14.42468719051704, −13.78095079043862, −13.01578957687602, −12.91435646634104, −12.16164041183359, −11.72951788068326, −11.08789550291803, −10.54805014074714, −10.19612068314513, −9.400619906100071, −8.929752548904583, −8.535660772013874, −7.695591386043909, −7.363223628213568, −6.579594633535588, −6.139068123567254, −5.559084202903886, −4.738741771110411, −4.233568993522513, −3.719147349592081, −2.821206403552496, −2.103070089897759, −1.610619752602737, −0.3107360053896557,
0.3107360053896557, 1.610619752602737, 2.103070089897759, 2.821206403552496, 3.719147349592081, 4.233568993522513, 4.738741771110411, 5.559084202903886, 6.139068123567254, 6.579594633535588, 7.363223628213568, 7.695591386043909, 8.535660772013874, 8.929752548904583, 9.400619906100071, 10.19612068314513, 10.54805014074714, 11.08789550291803, 11.72951788068326, 12.16164041183359, 12.91435646634104, 13.01578957687602, 13.78095079043862, 14.42468719051704, 14.77842230353687
