| L(s) = 1 | + (0.728 − 0.684i)2-s + (−1.52 − 2.40i)3-s + (0.0627 − 0.998i)4-s + (2.07 − 0.831i)5-s + (−2.75 − 0.708i)6-s + (0.278 − 0.858i)7-s + (−0.637 − 0.770i)8-s + (−2.17 + 4.62i)9-s + (0.943 − 2.02i)10-s + (−0.654 + 0.614i)11-s + (−2.49 + 1.37i)12-s + (0.895 − 1.90i)13-s + (−0.384 − 0.816i)14-s + (−5.16 − 3.72i)15-s + (−0.992 − 0.125i)16-s + (0.270 + 4.29i)17-s + ⋯ |
| L(s) = 1 | + (0.515 − 0.484i)2-s + (−0.880 − 1.38i)3-s + (0.0313 − 0.499i)4-s + (0.928 − 0.371i)5-s + (−1.12 − 0.289i)6-s + (0.105 − 0.324i)7-s + (−0.225 − 0.272i)8-s + (−0.725 + 1.54i)9-s + (0.298 − 0.641i)10-s + (−0.197 + 0.185i)11-s + (−0.720 + 0.396i)12-s + (0.248 − 0.527i)13-s + (−0.102 − 0.218i)14-s + (−1.33 − 0.960i)15-s + (−0.248 − 0.0313i)16-s + (0.0655 + 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.445821 - 1.28957i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.445821 - 1.28957i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.728 + 0.684i)T \) |
| 5 | \( 1 + (-2.07 + 0.831i)T \) |
| good | 3 | \( 1 + (1.52 + 2.40i)T + (-1.27 + 2.71i)T^{2} \) |
| 7 | \( 1 + (-0.278 + 0.858i)T + (-5.66 - 4.11i)T^{2} \) |
| 11 | \( 1 + (0.654 - 0.614i)T + (0.690 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.895 + 1.90i)T + (-8.28 - 10.0i)T^{2} \) |
| 17 | \( 1 + (-0.270 - 4.29i)T + (-16.8 + 2.13i)T^{2} \) |
| 19 | \( 1 + (3.27 - 5.16i)T + (-8.08 - 17.1i)T^{2} \) |
| 23 | \( 1 + (-1.10 + 5.79i)T + (-21.3 - 8.46i)T^{2} \) |
| 29 | \( 1 + (-4.40 - 1.74i)T + (21.1 + 19.8i)T^{2} \) |
| 31 | \( 1 + (0.385 + 6.12i)T + (-30.7 + 3.88i)T^{2} \) |
| 37 | \( 1 + (-2.61 - 0.329i)T + (35.8 + 9.20i)T^{2} \) |
| 41 | \( 1 + (0.0755 + 0.396i)T + (-38.1 + 15.0i)T^{2} \) |
| 43 | \( 1 + (-9.59 - 6.97i)T + (13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (-8.16 + 9.86i)T + (-8.80 - 46.1i)T^{2} \) |
| 53 | \( 1 + (3.27 - 0.839i)T + (46.4 - 25.5i)T^{2} \) |
| 59 | \( 1 + (6.89 - 3.79i)T + (31.6 - 49.8i)T^{2} \) |
| 61 | \( 1 + (1.12 - 5.89i)T + (-56.7 - 22.4i)T^{2} \) |
| 67 | \( 1 + (9.03 - 3.57i)T + (48.8 - 45.8i)T^{2} \) |
| 71 | \( 1 + (0.216 - 0.262i)T + (-13.3 - 69.7i)T^{2} \) |
| 73 | \( 1 + (9.36 + 5.14i)T + (39.1 + 61.6i)T^{2} \) |
| 79 | \( 1 + (2.70 + 4.25i)T + (-33.6 + 71.4i)T^{2} \) |
| 83 | \( 1 + (-4.17 + 6.58i)T + (-35.3 - 75.1i)T^{2} \) |
| 89 | \( 1 + (0.225 + 0.124i)T + (47.6 + 75.1i)T^{2} \) |
| 97 | \( 1 + (16.8 + 6.65i)T + (70.7 + 66.4i)T^{2} \) |
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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
−12.06141362711115672651177735601, −10.73447054417607160924465714931, −10.30344643739941156422843981417, −8.660471916849981422978926219534, −7.52759988867778611165983935286, −6.13160729943751846521024359942, −5.91662747467622577602900594286, −4.45617712975685362277610967268, −2.33122880273691753732738394201, −1.13452609057236685798256003017,
2.88221047150523358945471802747, 4.39493237930208139734169982337, 5.27200812242061387511005048251, 6.03915564935794426781198597413, 7.08927699369965935855865761453, 8.940825412951569671206324713007, 9.526196329942197449605880253240, 10.70793190634032359348022715491, 11.26413663748878736662365495893, 12.32238225560891494826028342001
