L(s) = 1 | + (−1.19 − 2.06i)2-s + (1.37 − 2.38i)3-s + (−1.85 + 3.20i)4-s + (0.491 + 0.850i)5-s − 6.57·6-s + (1.69 + 2.03i)7-s + 4.06·8-s + (−2.28 − 3.95i)9-s + (1.17 − 2.03i)10-s + (−0.293 + 0.509i)11-s + (5.09 + 8.82i)12-s + (2.17 − 5.93i)14-s + 2.70·15-s + (−1.15 − 1.99i)16-s + (3.22 − 5.58i)17-s + (−5.45 + 9.45i)18-s + ⋯ |
L(s) = 1 | + (−0.844 − 1.46i)2-s + (0.794 − 1.37i)3-s + (−0.925 + 1.60i)4-s + (0.219 + 0.380i)5-s − 2.68·6-s + (0.640 + 0.767i)7-s + 1.43·8-s + (−0.761 − 1.31i)9-s + (0.370 − 0.642i)10-s + (−0.0886 + 0.153i)11-s + (1.47 + 2.54i)12-s + (0.581 − 1.58i)14-s + 0.697·15-s + (−0.288 − 0.498i)16-s + (0.782 − 1.35i)17-s + (−1.28 + 2.22i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 - 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.915 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.272682220\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.272682220\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-1.69 - 2.03i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (1.19 + 2.06i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.37 + 2.38i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.491 - 0.850i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.293 - 0.509i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-3.22 + 5.58i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.91 + 3.31i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.13 + 7.15i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 3.96T + 29T^{2} \) |
| 31 | \( 1 + (1.49 - 2.58i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.877 - 1.52i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 3.67T + 41T^{2} \) |
| 43 | \( 1 - 6.38T + 43T^{2} \) |
| 47 | \( 1 + (2.17 + 3.75i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.212 - 0.368i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.00 + 5.20i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.10 + 1.91i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.50 + 6.07i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3.60T + 71T^{2} \) |
| 73 | \( 1 + (-2.46 + 4.27i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.39 + 2.41i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 2.86T + 83T^{2} \) |
| 89 | \( 1 + (1.04 + 1.81i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 7.69T + 97T^{2} \) |
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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
−9.209461526599998063665252823886, −8.592382294485326420182044795860, −8.001452961607083991764459399772, −7.18172489704164016407464454054, −6.20055535482824554517766441355, −4.76304515611060771165530975732, −3.23440885229328618974170815427, −2.43804046918812726096077969034, −2.00047283666254806856902486599, −0.68418433516236273367379210786,
1.54008603065230038616363744320, 3.56731546309962100460960668648, 4.26498697675235411183452722206, 5.38233496988131434761946333193, 5.88815083630141567121661269663, 7.31436628603161298387569202692, 7.973244232174603798253448483109, 8.488679842523830593103791053630, 9.283430735659107849599234672289, 9.931603273933466687194930516636