L(s) = 1 | − 1.80i·3-s + (1.73 + 1.41i)5-s − 2.12·7-s − 0.267·9-s − 4.11i·11-s + 2.72·13-s + (2.55 − 3.13i)15-s + 1.79i·17-s + (3.49 + 2.60i)19-s + 3.85i·21-s − 3.68·23-s + (0.999 + 4.89i)25-s − 4.93i·27-s − 10.5i·29-s + 4.42·31-s + ⋯ |
L(s) = 1 | − 1.04i·3-s + (0.774 + 0.632i)5-s − 0.804·7-s − 0.0893·9-s − 1.24i·11-s + 0.755·13-s + (0.660 − 0.808i)15-s + 0.434i·17-s + (0.801 + 0.598i)19-s + 0.840i·21-s − 0.769·23-s + (0.199 + 0.979i)25-s − 0.950i·27-s − 1.95i·29-s + 0.795·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.875164995\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.875164995\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.73 - 1.41i)T \) |
| 19 | \( 1 + (-3.49 - 2.60i)T \) |
good | 3 | \( 1 + 1.80iT - 3T^{2} \) |
| 7 | \( 1 + 2.12T + 7T^{2} \) |
| 11 | \( 1 + 4.11iT - 11T^{2} \) |
| 13 | \( 1 - 2.72T + 13T^{2} \) |
| 17 | \( 1 - 1.79iT - 17T^{2} \) |
| 23 | \( 1 + 3.68T + 23T^{2} \) |
| 29 | \( 1 + 10.5iT - 29T^{2} \) |
| 31 | \( 1 - 4.42T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 + 3.85iT - 41T^{2} \) |
| 43 | \( 1 + 7.94T + 43T^{2} \) |
| 47 | \( 1 - 6.38T + 47T^{2} \) |
| 53 | \( 1 + 4.71T + 53T^{2} \) |
| 59 | \( 1 + 4.42T + 59T^{2} \) |
| 61 | \( 1 - 8.19T + 61T^{2} \) |
| 67 | \( 1 + 3.13iT - 67T^{2} \) |
| 71 | \( 1 - 16.5T + 71T^{2} \) |
| 73 | \( 1 + 13.3iT - 73T^{2} \) |
| 79 | \( 1 + 6.29T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 - 10.5iT - 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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
−9.475720866843669962770407687992, −8.222539408207384931714442237999, −7.78491360482779670380498300096, −6.54228947948169840970286908926, −6.26683970505418616610138396683, −5.65399479862715130857305788482, −3.95690724966183086857343020627, −3.05565104451144876249603512317, −2.04904241998673656131623496445, −0.833636028936640232739233535069,
1.30864976419199691989807100399, 2.70904642172997030440259019366, 3.78798911379585341694070785094, 4.71302608104864912753465018821, 5.26009016554361286907481250273, 6.34929093809138011794768574187, 7.07504022003570817712334282578, 8.257066576262541920375186481915, 9.249069513654936502406165215824, 9.621211212215332945131746946693