L(s) = 1 | + (0.587 − 1.28i)2-s + (1.69 + 0.378i)3-s + (−1.30 − 1.51i)4-s + (−1.97 + 1.14i)5-s + (1.47 − 1.95i)6-s + (−0.907 + 1.57i)7-s + (−2.71 + 0.795i)8-s + (2.71 + 1.27i)9-s + (0.306 + 3.21i)10-s + (−4.24 − 2.44i)11-s + (−1.64 − 3.05i)12-s + (4.00 − 2.31i)13-s + (1.48 + 2.09i)14-s + (−3.77 + 1.18i)15-s + (−0.570 + 3.95i)16-s + 1.92·17-s + ⋯ |
L(s) = 1 | + (0.415 − 0.909i)2-s + (0.975 + 0.218i)3-s + (−0.654 − 0.755i)4-s + (−0.883 + 0.510i)5-s + (0.604 − 0.796i)6-s + (−0.343 + 0.594i)7-s + (−0.959 + 0.281i)8-s + (0.904 + 0.426i)9-s + (0.0968 + 1.01i)10-s + (−1.27 − 0.738i)11-s + (−0.473 − 0.880i)12-s + (1.11 − 0.641i)13-s + (0.398 + 0.559i)14-s + (−0.973 + 0.304i)15-s + (−0.142 + 0.989i)16-s + 0.467·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.07287 - 0.487486i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.07287 - 0.487486i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.587 + 1.28i)T \) |
| 3 | \( 1 + (-1.69 - 0.378i)T \) |
good | 5 | \( 1 + (1.97 - 1.14i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (0.907 - 1.57i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (4.24 + 2.44i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.00 + 2.31i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 1.92T + 17T^{2} \) |
| 19 | \( 1 - 2.12iT - 19T^{2} \) |
| 23 | \( 1 + (1.15 + 2.00i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.16 - 1.82i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.65 + 4.60i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 7.98iT - 37T^{2} \) |
| 41 | \( 1 + (2.36 + 4.09i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.20 + 1.27i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.02 - 3.49i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 8.95iT - 53T^{2} \) |
| 59 | \( 1 + (-3.05 + 1.76i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.71 - 0.991i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.72 - 4.46i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 + (4.97 - 8.61i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.12 - 1.80i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 2.49T + 89T^{2} \) |
| 97 | \( 1 + (-6.99 + 12.1i)T + (-48.5 - 84.0i)T^{2} \) |
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
−14.38617051101068374344458415225, −13.34373313837765158441049972729, −12.46950248130481375709006721605, −11.05199725887102069148058036778, −10.27357586118884874064143799284, −8.824375187601826531347937934966, −7.85702128582173897228789963487, −5.68312454052873255757110201877, −3.78299947695915591829500263405, −2.82368472877987340325548906309,
3.48748889216191682597875390263, 4.70070231611120625295778626878, 6.77717620251028244551546295518, 7.82605066229181761490246488452, 8.582089976265332008898049886723, 9.982080615838929807857780270381, 11.95254456046828480057570097692, 13.09422987969527482029447139767, 13.62865613062889630224978825829, 14.91802377515106952169637665344