| L(s) = 1 | + (0.642 − 0.136i)2-s + (−0.199 − 1.90i)3-s + (−1.43 + 0.638i)4-s + (−2.38 + 2.65i)5-s + (−0.388 − 1.19i)6-s + (2.59 − 0.531i)7-s + (−1.89 + 1.37i)8-s + (−0.642 + 0.136i)9-s + (−1.17 + 2.02i)10-s + (1.5 + 2.59i)12-s + (1.82 − 5.62i)13-s + (1.59 − 0.695i)14-s + (5.52 + 4.01i)15-s + (1.06 − 1.18i)16-s + (−1.62 − 0.344i)17-s + (−0.393 + 0.175i)18-s + ⋯ |
| L(s) = 1 | + (0.454 − 0.0965i)2-s + (−0.115 − 1.09i)3-s + (−0.716 + 0.319i)4-s + (−1.06 + 1.18i)5-s + (−0.158 − 0.487i)6-s + (0.979 − 0.200i)7-s + (−0.670 + 0.486i)8-s + (−0.214 + 0.0455i)9-s + (−0.370 + 0.641i)10-s + (0.433 + 0.749i)12-s + (0.506 − 1.55i)13-s + (0.425 − 0.185i)14-s + (1.42 + 1.03i)15-s + (0.267 − 0.297i)16-s + (−0.393 − 0.0835i)17-s + (−0.0928 + 0.0413i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.343 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.607403 - 0.868664i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.607403 - 0.868664i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.59 + 0.531i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (-0.642 + 0.136i)T + (1.82 - 0.813i)T^{2} \) |
| 3 | \( 1 + (0.199 + 1.90i)T + (-2.93 + 0.623i)T^{2} \) |
| 5 | \( 1 + (2.38 - 2.65i)T + (-0.522 - 4.97i)T^{2} \) |
| 13 | \( 1 + (-1.82 + 5.62i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (1.62 + 0.344i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (1.35 + 0.602i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (1.67 + 2.89i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.49 - 1.81i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (4.73 + 5.26i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (-0.471 + 4.48i)T + (-36.1 - 7.69i)T^{2} \) |
| 41 | \( 1 + (1.04 - 0.755i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 1.59T + 43T^{2} \) |
| 47 | \( 1 + (1.51 + 0.673i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-6.17 - 6.85i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (-8.08 + 3.60i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-4.47 + 4.96i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-4.91 + 8.50i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.66 + 8.19i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (4.16 - 1.85i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (6.25 - 1.32i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (0.0518 + 0.159i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-1.28 - 2.22i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.00 + 9.26i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.23080790154189576525654547747, −8.681589313830857087169095483045, −7.933580021146769712851306530544, −7.57029124211007817799847813317, −6.58846911436336385387080503100, −5.53939794233121485626530194023, −4.33813303928676651055929582061, −3.51598183340905965503049685809, −2.39639167287455949805667972491, −0.50648693361239795889756891278,
1.43482850962966686998020647009, 3.85963926816359325425552990827, 4.20027028187517868557459622038, 4.87194486824925899522383780023, 5.54125448437949218180608838160, 6.98549970397788630947024550199, 8.373509262388087712078056901808, 8.721429882584468631448819794762, 9.452613383132547655752456872812, 10.39695793939965542695772471473
