L(s) = 1 | + (−0.965 + 0.258i)2-s + (−1.73 − 0.0795i)3-s + (0.866 − 0.499i)4-s + (−1.51 − 1.64i)5-s + (1.69 − 0.370i)6-s + (−1.00 − 3.75i)7-s + (−0.707 + 0.707i)8-s + (2.98 + 0.275i)9-s + (1.89 + 1.19i)10-s + (−3.44 − 1.98i)11-s + (−1.53 + 0.796i)12-s + (−0.256 + 0.956i)13-s + (1.94 + 3.36i)14-s + (2.49 + 2.95i)15-s + (0.500 − 0.866i)16-s + (0.120 + 0.120i)17-s + ⋯ |
L(s) = 1 | + (−0.683 + 0.183i)2-s + (−0.998 − 0.0459i)3-s + (0.433 − 0.249i)4-s + (−0.679 − 0.733i)5-s + (0.690 − 0.151i)6-s + (−0.380 − 1.41i)7-s + (−0.249 + 0.249i)8-s + (0.995 + 0.0917i)9-s + (0.598 + 0.376i)10-s + (−1.03 − 0.599i)11-s + (−0.444 + 0.229i)12-s + (−0.0710 + 0.265i)13-s + (0.519 + 0.899i)14-s + (0.644 + 0.764i)15-s + (0.125 − 0.216i)16-s + (0.0291 + 0.0291i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.328 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.208610 - 0.293496i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.208610 - 0.293496i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 - 0.258i)T \) |
| 3 | \( 1 + (1.73 + 0.0795i)T \) |
| 5 | \( 1 + (1.51 + 1.64i)T \) |
good | 7 | \( 1 + (1.00 + 3.75i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (3.44 + 1.98i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.256 - 0.956i)T + (-11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (-0.120 - 0.120i)T + 17iT^{2} \) |
| 19 | \( 1 - 1.88iT - 19T^{2} \) |
| 23 | \( 1 + (-5.08 - 1.36i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-2.15 + 3.73i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4.70 + 8.14i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.26 + 3.26i)T - 37iT^{2} \) |
| 41 | \( 1 + (-7.15 + 4.13i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.99 - 0.533i)T + (37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (-3.34 + 0.897i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-3.66 + 3.66i)T - 53iT^{2} \) |
| 59 | \( 1 + (2.72 + 4.72i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.35 - 7.54i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.86 - 2.10i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 6.94iT - 71T^{2} \) |
| 73 | \( 1 + (8.27 + 8.27i)T + 73iT^{2} \) |
| 79 | \( 1 + (-11.7 - 6.78i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.81 - 6.75i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + 4.87T + 89T^{2} \) |
| 97 | \( 1 + (0.387 + 1.44i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−13.51794754817995718839969461261, −12.68638118783345156022967675654, −11.34932792719517834616561593587, −10.66616038458908482950589325647, −9.536860956779032308503207368132, −7.921173575873059110492692729208, −7.13271460238210307692755659808, −5.61077309093021410393670229855, −4.11900219054840968756339190750, −0.61006501612829189267912984801,
2.81925653112444433162049505195, 5.09261905803672684379667766706, 6.50721842454296087605454564206, 7.59382230381229383766826271875, 9.049382547311596331398043764727, 10.34058356079336317102549784173, 11.10360773691629744994584167634, 12.18560624158737866481894388966, 12.79309291304006973236648691738, 14.96472441509450193790184907452