| L(s) = 1 | − 2-s + 4-s − 8-s − 4·11-s − 13-s + 16-s − 3·17-s + 2·19-s + 4·22-s − 23-s + 26-s − 8·29-s + 2·31-s − 32-s + 3·34-s + 8·37-s − 2·38-s + 6·41-s + 4·43-s − 4·44-s + 46-s + 7·47-s − 52-s + 6·53-s + 8·58-s − 5·59-s + 10·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.20·11-s − 0.277·13-s + 1/4·16-s − 0.727·17-s + 0.458·19-s + 0.852·22-s − 0.208·23-s + 0.196·26-s − 1.48·29-s + 0.359·31-s − 0.176·32-s + 0.514·34-s + 1.31·37-s − 0.324·38-s + 0.937·41-s + 0.609·43-s − 0.603·44-s + 0.147·46-s + 1.02·47-s − 0.138·52-s + 0.824·53-s + 1.05·58-s − 0.650·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.528987751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.528987751\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
| 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
−12.68635398751385, −12.32737515138986, −11.53647363096102, −11.37205943915212, −10.87586626152852, −10.33873447887486, −10.03014058690871, −9.488803822339978, −8.974241941489185, −8.729683293235597, −7.921681067941542, −7.574253552735311, −7.449800779302247, −6.672037645222807, −6.142870782494522, −5.663088042200307, −5.223595739163153, −4.592507705176470, −4.038034862374013, −3.440648102101685, −2.691967864403985, −2.346929696790205, −1.877342205406113, −0.8709519721899495, −0.4610707266975014,
0.4610707266975014, 0.8709519721899495, 1.877342205406113, 2.346929696790205, 2.691967864403985, 3.440648102101685, 4.038034862374013, 4.592507705176470, 5.223595739163153, 5.663088042200307, 6.142870782494522, 6.672037645222807, 7.449800779302247, 7.574253552735311, 7.921681067941542, 8.729683293235597, 8.974241941489185, 9.488803822339978, 10.03014058690871, 10.33873447887486, 10.87586626152852, 11.37205943915212, 11.53647363096102, 12.32737515138986, 12.68635398751385
