| L(s) = 1 | + (0.654 − 0.755i)2-s + (0.841 − 0.540i)3-s + (−0.142 − 0.989i)4-s + (−1.38 − 3.02i)5-s + (0.142 − 0.989i)6-s + (−0.959 + 0.281i)7-s + (−0.841 − 0.540i)8-s + (0.415 − 0.909i)9-s + (−3.19 − 0.938i)10-s + (−0.142 − 0.164i)11-s + (−0.654 − 0.755i)12-s + (−0.305 − 0.0898i)13-s + (−0.415 + 0.909i)14-s + (−2.80 − 1.80i)15-s + (−0.959 + 0.281i)16-s + (−0.409 + 2.85i)17-s + ⋯ |
| L(s) = 1 | + (0.463 − 0.534i)2-s + (0.485 − 0.312i)3-s + (−0.0711 − 0.494i)4-s + (−0.618 − 1.35i)5-s + (0.0580 − 0.404i)6-s + (−0.362 + 0.106i)7-s + (−0.297 − 0.191i)8-s + (0.138 − 0.303i)9-s + (−1.01 − 0.296i)10-s + (−0.0430 − 0.0496i)11-s + (−0.189 − 0.218i)12-s + (−0.0848 − 0.0249i)13-s + (−0.111 + 0.243i)14-s + (−0.723 − 0.464i)15-s + (−0.239 + 0.0704i)16-s + (−0.0994 + 0.691i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0244i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0244i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0185487 + 1.51803i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0185487 + 1.51803i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.654 + 0.755i)T \) |
| 3 | \( 1 + (-0.841 + 0.540i)T \) |
| 7 | \( 1 + (0.959 - 0.281i)T \) |
| 23 | \( 1 + (4.75 - 0.622i)T \) |
| good | 5 | \( 1 + (1.38 + 3.02i)T + (-3.27 + 3.77i)T^{2} \) |
| 11 | \( 1 + (0.142 + 0.164i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.305 + 0.0898i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.409 - 2.85i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (0.837 + 5.82i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-1.18 + 8.27i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-7.89 - 5.07i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (3.22 - 7.06i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (4.65 + 10.1i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (6.45 - 4.15i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + 2.06T + 47T^{2} \) |
| 53 | \( 1 + (-0.751 + 0.220i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-4.56 - 1.34i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-12.5 - 8.04i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-2.78 + 3.21i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-5.68 + 6.55i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (1.19 + 8.30i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (11.0 + 3.25i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-1.66 + 3.65i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-9.93 + 6.38i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (4.00 + 8.75i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−9.634360731406477012827996068147, −8.483034050926286672837089127024, −8.408101360563994993753044580739, −7.03713074017218288041835167307, −6.05762697180711233442148465655, −4.89974001714520707813091848978, −4.24702470296504824008423685746, −3.19641541068160603193642582879, −1.93607280047032851631236882199, −0.55271545951972683719335905513,
2.40905168999453306134167830482, 3.40489864508624239361719114211, 3.98235731609963050496372801548, 5.21453753523747299514677850418, 6.41116628358709926312892438839, 6.96707365352835995389705824805, 7.86001193829134083437849007419, 8.468956088937947894278580493660, 9.803813681185652262094533279432, 10.26520480924217850725359892823
