L(s) = 1 | + (−0.866 + 1.5i)3-s + (1.73 − i)5-s + (2 − 3.46i)7-s + (−1.5 − 2.59i)9-s + (2.59 + 1.5i)11-s + (−1.73 + i)13-s + 3.46i·15-s + 5·17-s + i·19-s + (3.46 + 6i)21-s + (1 + 1.73i)23-s + (−0.500 + 0.866i)25-s + 5.19·27-s + (−2 − 3.46i)31-s + (−4.5 + 2.59i)33-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.866i)3-s + (0.774 − 0.447i)5-s + (0.755 − 1.30i)7-s + (−0.5 − 0.866i)9-s + (0.783 + 0.452i)11-s + (−0.480 + 0.277i)13-s + 0.894i·15-s + 1.21·17-s + 0.229i·19-s + (0.755 + 1.30i)21-s + (0.208 + 0.361i)23-s + (−0.100 + 0.173i)25-s + 1.00·27-s + (−0.359 − 0.622i)31-s + (−0.783 + 0.452i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.34312 + 0.0586421i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.34312 + 0.0586421i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.866 - 1.5i)T \) |
good | 5 | \( 1 + (-1.73 + i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-2 + 3.46i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.59 - 1.5i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.73 - i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 19 | \( 1 - iT - 19T^{2} \) |
| 23 | \( 1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + (-2.5 - 4.33i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (9.52 + 5.5i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + (0.866 - 0.5i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (10.3 + 6i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.59 + 1.5i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 9T + 73T^{2} \) |
| 79 | \( 1 + (7 - 12.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.46 - 2i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T + (-48.5 - 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.65962735170445954671029139697, −10.82361635479497916399639322249, −9.828816828054340647931471216438, −9.431596047719078711351938177636, −7.980450472901583478422382616210, −6.84576408955375056866623468585, −5.57972333927483068417834452888, −4.68312607265359988240416812973, −3.70463821838737738958723908868, −1.38447067023982624408387265594,
1.65162219993007474554059115777, 2.85041523031160849579831399260, 5.10559013535950030807782824660, 5.83158550300280307657623435153, 6.71076027218495004892389958893, 7.941798803842492948921589613286, 8.817848463775925318857141361342, 9.985315447175374822112985958760, 11.07557443876400688469687811823, 11.92929106733060566275595330594