L(s) = 1 | + (0.327 − 1.37i)2-s + (−0.844 + 1.51i)3-s + (−1.78 − 0.900i)4-s + (−0.891 − 3.32i)5-s + (1.80 + 1.65i)6-s + (3.95 − 2.28i)7-s + (−1.82 + 2.16i)8-s + (−1.57 − 2.55i)9-s + (−4.86 + 0.138i)10-s + (−2.12 − 0.568i)11-s + (2.86 − 1.93i)12-s + (−0.0649 + 0.0174i)13-s + (−1.84 − 6.19i)14-s + (5.78 + 1.46i)15-s + (2.37 + 3.21i)16-s − 0.00952·17-s + ⋯ |
L(s) = 1 | + (0.231 − 0.972i)2-s + (−0.487 + 0.872i)3-s + (−0.892 − 0.450i)4-s + (−0.398 − 1.48i)5-s + (0.736 + 0.676i)6-s + (1.49 − 0.863i)7-s + (−0.644 + 0.764i)8-s + (−0.524 − 0.851i)9-s + (−1.53 + 0.0436i)10-s + (−0.639 − 0.171i)11-s + (0.828 − 0.560i)12-s + (−0.0180 + 0.00482i)13-s + (−0.494 − 1.65i)14-s + (1.49 + 0.377i)15-s + (0.594 + 0.803i)16-s − 0.00231·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.559272 - 0.789690i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.559272 - 0.789690i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.327 + 1.37i)T \) |
| 3 | \( 1 + (0.844 - 1.51i)T \) |
good | 5 | \( 1 + (0.891 + 3.32i)T + (-4.33 + 2.5i)T^{2} \) |
| 7 | \( 1 + (-3.95 + 2.28i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.12 + 0.568i)T + (9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (0.0649 - 0.0174i)T + (11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + 0.00952T + 17T^{2} \) |
| 19 | \( 1 + (-2.79 - 2.79i)T + 19iT^{2} \) |
| 23 | \( 1 + (-4.81 - 2.78i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.343 + 1.28i)T + (-25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (1.30 - 2.25i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.32 - 3.32i)T - 37iT^{2} \) |
| 41 | \( 1 + (-4.34 - 2.50i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.68 - 0.718i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (2.77 + 4.81i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.80 + 1.80i)T - 53iT^{2} \) |
| 59 | \( 1 + (0.0592 + 0.221i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-2.91 + 10.8i)T + (-52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (4.45 - 1.19i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 7.21iT - 71T^{2} \) |
| 73 | \( 1 + 6.99iT - 73T^{2} \) |
| 79 | \( 1 + (-6.86 - 11.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.18 - 11.8i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + 2.28iT - 89T^{2} \) |
| 97 | \( 1 + (4.69 + 8.12i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.55198076466425639386292441224, −11.60930814740281566171771191075, −10.98515403907098885672131884114, −9.943425387695028702486775284732, −8.819714161460952751145479237650, −7.944817931169800214724034376334, −5.29047315475895443739875398030, −4.86048600514155481089032030927, −3.79792940350450381479372950457, −1.10197584782988246405768567724,
2.68695137991744985013307939998, 4.87095813468885882874418321851, 5.89179396887739854070981951244, 7.16102102881686275791083270737, 7.67209721273952001643460406559, 8.781781359317048989946230153179, 10.70584942880261165704515395995, 11.46901051987054610990211131570, 12.43064313506600407522059297528, 13.64525243455214085019618057383