| L(s) = 1 | + (−0.597 + 1.36i)2-s + (−0.159 − 0.171i)4-s + (3.23 + 0.487i)5-s + (3.25 + 3.02i)7-s + (−2.49 + 0.871i)8-s + (−2.59 + 4.13i)10-s + (−2.45 + 2.10i)11-s + (0.198 − 0.291i)13-s + (−6.08 + 2.65i)14-s + (0.329 − 4.40i)16-s + (0.295 + 0.295i)17-s + (−2.84 − 1.78i)19-s + (−0.430 − 0.631i)20-s + (−1.42 − 4.61i)22-s + (3.93 + 1.54i)23-s + ⋯ |
| L(s) = 1 | + (−0.422 + 0.968i)2-s + (−0.0795 − 0.0857i)4-s + (1.44 + 0.217i)5-s + (1.23 + 1.14i)7-s + (−0.880 + 0.308i)8-s + (−0.822 + 1.30i)10-s + (−0.738 + 0.635i)11-s + (0.0551 − 0.0808i)13-s + (−1.62 + 0.709i)14-s + (0.0824 − 1.10i)16-s + (0.0716 + 0.0716i)17-s + (−0.651 − 0.409i)19-s + (−0.0963 − 0.141i)20-s + (−0.303 − 0.984i)22-s + (0.820 + 0.321i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.660110 + 1.60492i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.660110 + 1.60492i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 + (1.14 + 5.26i)T \) |
| good | 2 | \( 1 + (0.597 - 1.36i)T + (-1.36 - 1.46i)T^{2} \) |
| 5 | \( 1 + (-3.23 - 0.487i)T + (4.77 + 1.47i)T^{2} \) |
| 7 | \( 1 + (-3.25 - 3.02i)T + (0.523 + 6.98i)T^{2} \) |
| 11 | \( 1 + (2.45 - 2.10i)T + (1.63 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.198 + 0.291i)T + (-4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (-0.295 - 0.295i)T + 17iT^{2} \) |
| 19 | \( 1 + (2.84 + 1.78i)T + (8.24 + 17.1i)T^{2} \) |
| 23 | \( 1 + (-3.93 - 1.54i)T + (16.8 + 15.6i)T^{2} \) |
| 31 | \( 1 + (1.57 - 1.16i)T + (9.13 - 29.6i)T^{2} \) |
| 37 | \( 1 + (2.56 + 7.33i)T + (-28.9 + 23.0i)T^{2} \) |
| 41 | \( 1 + (-10.1 - 2.72i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-5.81 + 7.88i)T + (-12.6 - 41.0i)T^{2} \) |
| 47 | \( 1 + (-5.33 - 6.20i)T + (-7.00 + 46.4i)T^{2} \) |
| 53 | \( 1 + (9.02 + 7.19i)T + (11.7 + 51.6i)T^{2} \) |
| 59 | \( 1 + (8.26 - 4.77i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.88 - 0.182i)T + (60.8 - 4.55i)T^{2} \) |
| 67 | \( 1 + (5.22 - 0.391i)T + (66.2 - 9.98i)T^{2} \) |
| 71 | \( 1 + (-3.35 + 1.61i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-0.101 + 0.904i)T + (-71.1 - 16.2i)T^{2} \) |
| 79 | \( 1 + (0.675 - 3.57i)T + (-73.5 - 28.8i)T^{2} \) |
| 83 | \( 1 + (-2.44 + 7.94i)T + (-68.5 - 46.7i)T^{2} \) |
| 89 | \( 1 + (7.60 - 0.857i)T + (86.7 - 19.8i)T^{2} \) |
| 97 | \( 1 + (4.57 + 8.65i)T + (-54.6 + 80.1i)T^{2} \) |
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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
−10.54893570067640702683257999931, −9.292737870332886156474068868801, −9.020708056399897470060304127296, −7.948236099438919908289201682444, −7.28581274481670787898881576264, −6.07103909588652696121771032076, −5.65286237855508551392810276741, −4.77786484738228134127019998847, −2.64364021484959908311989649341, −2.00847321533946047822500438646,
1.05487945817002419558134809342, 1.88494088024502096680998969944, 3.01627907481544809287143750513, 4.48577616726994388548852267773, 5.50089858126635976763833760645, 6.34568618056287075819563741697, 7.53301703988088599744812418007, 8.563412859995478223940659625311, 9.337972224891316982876216336014, 10.20314656973646146734788237135
