| L(s) = 1 | + (0.742 + 0.317i)2-s + (−0.932 − 0.974i)4-s + (−0.618 − 0.200i)5-s + (−0.732 − 0.837i)7-s + (−0.949 − 2.53i)8-s + (−0.394 − 0.345i)10-s + (−2.86 + 0.789i)11-s + (−0.525 + 1.90i)13-s + (−0.277 − 0.853i)14-s + (−0.0231 + 0.516i)16-s + (−5.70 + 4.14i)17-s + (−1.80 − 3.36i)19-s + (0.380 + 0.789i)20-s + (−2.37 − 0.321i)22-s + (−0.286 + 1.25i)23-s + ⋯ |
| L(s) = 1 | + (0.524 + 0.224i)2-s + (−0.466 − 0.487i)4-s + (−0.276 − 0.0898i)5-s + (−0.276 − 0.316i)7-s + (−0.335 − 0.894i)8-s + (−0.124 − 0.109i)10-s + (−0.862 + 0.238i)11-s + (−0.145 + 0.528i)13-s + (−0.0741 − 0.228i)14-s + (−0.00579 + 0.129i)16-s + (−1.38 + 1.00i)17-s + (−0.414 − 0.771i)19-s + (0.0850 + 0.176i)20-s + (−0.506 − 0.0685i)22-s + (−0.0598 + 0.262i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.272i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0477129 - 0.343334i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0477129 - 0.343334i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 71 | \( 1 + (7.37 - 4.07i)T \) |
| good | 2 | \( 1 + (-0.742 - 0.317i)T + (1.38 + 1.44i)T^{2} \) |
| 5 | \( 1 + (0.618 + 0.200i)T + (4.04 + 2.93i)T^{2} \) |
| 7 | \( 1 + (0.732 + 0.837i)T + (-0.939 + 6.93i)T^{2} \) |
| 11 | \( 1 + (2.86 - 0.789i)T + (9.44 - 5.64i)T^{2} \) |
| 13 | \( 1 + (0.525 - 1.90i)T + (-11.1 - 6.66i)T^{2} \) |
| 17 | \( 1 + (5.70 - 4.14i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (1.80 + 3.36i)T + (-10.4 + 15.8i)T^{2} \) |
| 23 | \( 1 + (0.286 - 1.25i)T + (-20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (-1.89 + 0.256i)T + (27.9 - 7.71i)T^{2} \) |
| 31 | \( 1 + (-0.233 + 0.0104i)T + (30.8 - 2.77i)T^{2} \) |
| 37 | \( 1 + (2.57 + 11.2i)T + (-33.3 + 16.0i)T^{2} \) |
| 41 | \( 1 + (3.04 + 3.81i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (-2.14 - 0.193i)T + (42.3 + 7.67i)T^{2} \) |
| 47 | \( 1 + (-6.34 + 1.15i)T + (44.0 - 16.5i)T^{2} \) |
| 53 | \( 1 + (4.52 - 4.72i)T + (-2.37 - 52.9i)T^{2} \) |
| 59 | \( 1 + (-2.91 - 4.41i)T + (-23.1 + 54.2i)T^{2} \) |
| 61 | \( 1 + (-5.28 + 6.05i)T + (-8.18 - 60.4i)T^{2} \) |
| 67 | \( 1 + (8.38 - 8.01i)T + (3.00 - 66.9i)T^{2} \) |
| 73 | \( 1 + (-0.351 + 0.823i)T + (-50.4 - 52.7i)T^{2} \) |
| 79 | \( 1 + (-10.8 + 4.08i)T + (59.4 - 51.9i)T^{2} \) |
| 83 | \( 1 + (-4.13 + 2.73i)T + (32.6 - 76.3i)T^{2} \) |
| 89 | \( 1 + (2.51 + 2.40i)T + (3.99 + 88.9i)T^{2} \) |
| 97 | \( 1 + (-9.12 - 7.27i)T + (21.5 + 94.5i)T^{2} \) |
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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
−10.31618063020823065704855115400, −9.251110346831629292206543776253, −8.566414180565786279381900421387, −7.32917061380387529722058678099, −6.50003580193634986324884238556, −5.56142798245406517964888734883, −4.49339488074025948100999270257, −3.88936481502967371015025051091, −2.19851205422989655780998998091, −0.14947802584548254536836057872,
2.46592160944871860865335164219, 3.31964309756013666546726893331, 4.49511152934861727732808217389, 5.28051006090652839508600790424, 6.38522340222834988514502296589, 7.62490295867102046793511292264, 8.322576847234203828413865464189, 9.155947111046340798872731739387, 10.16976624824625188151995022783, 11.14986585500487007372016936435
