L(s) = 1 | + (−0.755 + 0.947i)2-s + (0.215 + 1.43i)3-s + (0.118 + 0.517i)4-s + (1.58 + 2.33i)5-s + (−1.51 − 0.877i)6-s + (−1.01 + 0.584i)7-s + (−2.76 − 1.33i)8-s + (0.863 − 0.266i)9-s + (−3.40 − 0.255i)10-s + (−4.33 − 0.989i)11-s + (−0.715 + 0.280i)12-s + (0.132 + 1.76i)13-s + (0.211 − 1.40i)14-s + (−2.99 + 2.77i)15-s + (2.39 − 1.15i)16-s + (−2.23 − 3.46i)17-s + ⋯ |
L(s) = 1 | + (−0.534 + 0.670i)2-s + (0.124 + 0.826i)3-s + (0.0590 + 0.258i)4-s + (0.710 + 1.04i)5-s + (−0.620 − 0.358i)6-s + (−0.382 + 0.221i)7-s + (−0.977 − 0.470i)8-s + (0.287 − 0.0887i)9-s + (−1.07 − 0.0808i)10-s + (−1.30 − 0.298i)11-s + (−0.206 + 0.0811i)12-s + (0.0367 + 0.490i)13-s + (0.0564 − 0.374i)14-s + (−0.773 + 0.717i)15-s + (0.598 − 0.288i)16-s + (−0.541 − 0.840i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.332621 - 0.809858i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.332621 - 0.809858i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (2.23 + 3.46i)T \) |
| 43 | \( 1 + (-3.21 - 5.71i)T \) |
good | 2 | \( 1 + (0.755 - 0.947i)T + (-0.445 - 1.94i)T^{2} \) |
| 3 | \( 1 + (-0.215 - 1.43i)T + (-2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (-1.58 - 2.33i)T + (-1.82 + 4.65i)T^{2} \) |
| 7 | \( 1 + (1.01 - 0.584i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (4.33 + 0.989i)T + (9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (-0.132 - 1.76i)T + (-12.8 + 1.93i)T^{2} \) |
| 19 | \( 1 + (3.67 + 1.13i)T + (15.6 + 10.7i)T^{2} \) |
| 23 | \( 1 + (-1.03 + 1.11i)T + (-1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (1.31 - 8.69i)T + (-27.7 - 8.54i)T^{2} \) |
| 31 | \( 1 + (5.10 - 2.00i)T + (22.7 - 21.0i)T^{2} \) |
| 37 | \( 1 + (-8.49 - 4.90i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.60 - 2.87i)T + (9.12 + 39.9i)T^{2} \) |
| 47 | \( 1 + (2.29 + 10.0i)T + (-42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (0.241 - 3.21i)T + (-52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (-9.16 + 4.41i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (-4.01 - 1.57i)T + (44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (7.09 + 2.18i)T + (55.3 + 37.7i)T^{2} \) |
| 71 | \( 1 + (-4.06 - 4.37i)T + (-5.30 + 70.8i)T^{2} \) |
| 73 | \( 1 + (13.3 - 0.997i)T + (72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (0.790 - 0.456i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (15.5 - 2.34i)T + (79.3 - 24.4i)T^{2} \) |
| 89 | \( 1 + (1.67 - 0.252i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 + (-7.31 - 1.66i)T + (87.3 + 42.0i)T^{2} \) |
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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
−10.67482476482747251654229882005, −9.913129619276335714939384642729, −9.236984036171040746824266313191, −8.492648701175937601802472948144, −7.26297582945691955568781653016, −6.75832706001130333596755062288, −5.82725017147319815031261663274, −4.62573482775007302421672261588, −3.25401728933609471397948886735, −2.56090176791810162112870094253,
0.49683181150578865905427633522, 1.79987007657148121237734999797, 2.46369523268912641353278918557, 4.30264728943556131014286330801, 5.57728141612799831215525810250, 6.13679188063705124697075509237, 7.46045837130984182218387702986, 8.259552645912190493588645621470, 9.121237859876501644335682427481, 9.968406825586412575014274913770