| L(s) = 1 | + (−0.959 − 0.281i)2-s + (−0.755 − 0.654i)3-s + (0.841 + 0.540i)4-s + (−0.480 + 3.34i)5-s + (0.540 + 0.841i)6-s + (−0.131 + 2.64i)7-s + (−0.654 − 0.755i)8-s + (0.142 + 0.989i)9-s + (1.40 − 3.07i)10-s + (−0.0493 − 0.168i)11-s + (−0.281 − 0.959i)12-s + (2.65 + 1.21i)13-s + (0.870 − 2.49i)14-s + (2.55 − 2.21i)15-s + (0.415 + 0.909i)16-s + (−0.603 + 0.387i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 − 0.199i)2-s + (−0.436 − 0.378i)3-s + (0.420 + 0.270i)4-s + (−0.214 + 1.49i)5-s + (0.220 + 0.343i)6-s + (−0.0497 + 0.998i)7-s + (−0.231 − 0.267i)8-s + (0.0474 + 0.329i)9-s + (0.443 − 0.970i)10-s + (−0.0148 − 0.0506i)11-s + (−0.0813 − 0.276i)12-s + (0.735 + 0.335i)13-s + (0.232 − 0.667i)14-s + (0.658 − 0.570i)15-s + (0.103 + 0.227i)16-s + (−0.146 + 0.0940i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.224062 + 0.600933i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.224062 + 0.600933i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 + 0.281i)T \) |
| 3 | \( 1 + (0.755 + 0.654i)T \) |
| 7 | \( 1 + (0.131 - 2.64i)T \) |
| 23 | \( 1 + (3.94 - 2.72i)T \) |
| good | 5 | \( 1 + (0.480 - 3.34i)T + (-4.79 - 1.40i)T^{2} \) |
| 11 | \( 1 + (0.0493 + 0.168i)T + (-9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-2.65 - 1.21i)T + (8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (0.603 - 0.387i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-3.02 - 1.94i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-1.49 + 0.962i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-3.42 + 2.97i)T + (4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (7.95 - 1.14i)T + (35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (7.90 + 1.13i)T + (39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-0.300 - 0.260i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 1.90iT - 47T^{2} \) |
| 53 | \( 1 + (4.39 - 2.00i)T + (34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (0.693 + 0.316i)T + (38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-8.23 - 9.50i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-2.74 + 9.34i)T + (-56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (0.959 + 0.281i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (4.71 - 7.34i)T + (-30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (0.704 + 0.321i)T + (51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.697 - 4.85i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (6.55 - 7.56i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (1.81 - 12.6i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−10.35462307968685502366287454198, −9.690377426949280457280419683690, −8.582215760584834044678697812036, −7.86099332259458663657203972762, −6.90640573268667817019957892511, −6.31875067516431375070616096218, −5.47529924812019441511532249453, −3.75547144655046473832734218161, −2.81048089208100287315911310833, −1.75076548926961908227615197066,
0.41597609040673264932152515153, 1.42804404497727762014597369479, 3.47587725795516948213087969746, 4.55336288040066003049845087927, 5.23206458825282290782151987342, 6.32930278974626866756281574133, 7.26011873103403992944414400573, 8.293825902928421185179005444424, 8.730708411673408288453577612776, 9.755639315424551343536952247513
