| L(s) = 1 | + (−1.83 − 2.03i)2-s + (−0.676 − 1.59i)3-s + (−0.578 + 5.49i)4-s + (−2.11 − 0.733i)5-s + (−2.00 + 4.30i)6-s + (−0.931 + 1.61i)7-s + (7.83 − 5.69i)8-s + (−2.08 + 2.15i)9-s + (2.38 + 5.65i)10-s + (1.84 + 2.05i)11-s + (9.15 − 2.79i)12-s + (1.13 − 1.25i)13-s + (4.99 − 1.06i)14-s + (0.258 + 3.86i)15-s + (−15.1 − 3.22i)16-s + (−5.11 + 3.71i)17-s + ⋯ |
| L(s) = 1 | + (−1.29 − 1.44i)2-s + (−0.390 − 0.920i)3-s + (−0.289 + 2.74i)4-s + (−0.944 − 0.328i)5-s + (−0.820 + 1.75i)6-s + (−0.351 + 0.609i)7-s + (2.77 − 2.01i)8-s + (−0.694 + 0.719i)9-s + (0.753 + 1.78i)10-s + (0.557 + 0.619i)11-s + (2.64 − 0.807i)12-s + (0.314 − 0.349i)13-s + (1.33 − 0.283i)14-s + (0.0668 + 0.997i)15-s + (−3.79 − 0.806i)16-s + (−1.24 + 0.901i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.105i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.213739 + 0.0113187i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.213739 + 0.0113187i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.676 + 1.59i)T \) |
| 5 | \( 1 + (2.11 + 0.733i)T \) |
| good | 2 | \( 1 + (1.83 + 2.03i)T + (-0.209 + 1.98i)T^{2} \) |
| 7 | \( 1 + (0.931 - 1.61i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.84 - 2.05i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-1.13 + 1.25i)T + (-1.35 - 12.9i)T^{2} \) |
| 17 | \( 1 + (5.11 - 3.71i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.23 + 0.896i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (0.478 - 0.101i)T + (21.0 - 9.35i)T^{2} \) |
| 29 | \( 1 + (-1.65 + 0.735i)T + (19.4 - 21.5i)T^{2} \) |
| 31 | \( 1 + (2.49 + 1.11i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (-1.30 - 4.01i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (2.54 - 2.82i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (5.30 - 9.19i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.86 - 2.60i)T + (31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (-4.80 - 3.49i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (6.79 - 7.54i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-4.87 - 5.41i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (7.98 + 3.55i)T + (44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (-2.74 - 1.99i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.80 + 14.7i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (11.7 - 5.21i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-0.0919 - 0.874i)T + (-81.1 + 17.2i)T^{2} \) |
| 89 | \( 1 + (-1.51 + 4.65i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (3.46 - 1.54i)T + (64.9 - 72.0i)T^{2} \) |
| 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
−11.96752944890936538081304160604, −11.44468027992415157157325152275, −10.56222235980425029372643777533, −9.255720887576348898272986934615, −8.481170192044228472168115284269, −7.70901011847533521531757264347, −6.58660192618196954908504768946, −4.38662591603033974382489398609, −2.91935591819712378932352683975, −1.47285629734893521279024375222,
0.30038142615505558601513879804, 3.91676834719853299329559587144, 5.17576616540943916311160825976, 6.52073298011553093733195337398, 7.07366955221339167355362126476, 8.423326238801896657685791727847, 9.077846729030484864701608705819, 10.08522136956399746067187931793, 10.93336781490706319022717218494, 11.60192221363733605000870556023
