L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−2.61 + 0.397i)7-s + 0.999·8-s + (−0.429 − 0.248i)11-s + 2.74·13-s + (1.65 + 2.06i)14-s + (−0.5 − 0.866i)16-s + (3.16 + 1.82i)17-s + (−3.12 + 1.80i)19-s + 0.496i·22-s + (−3.21 − 5.56i)23-s + (−1.37 − 2.37i)26-s + (0.963 − 2.46i)28-s + 8.87i·29-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.988 + 0.150i)7-s + 0.353·8-s + (−0.129 − 0.0748i)11-s + 0.761·13-s + (0.441 + 0.552i)14-s + (−0.125 − 0.216i)16-s + (0.768 + 0.443i)17-s + (−0.716 + 0.413i)19-s + 0.105i·22-s + (−0.669 − 1.16i)23-s + (−0.269 − 0.466i)26-s + (0.182 − 0.465i)28-s + 1.64i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.796 + 0.604i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.796 + 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6479494460\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6479494460\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.61 - 0.397i)T \) |
good | 11 | \( 1 + (0.429 + 0.248i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2.74T + 13T^{2} \) |
| 17 | \( 1 + (-3.16 - 1.82i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.12 - 1.80i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.21 + 5.56i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 8.87iT - 29T^{2} \) |
| 31 | \( 1 + (6.90 + 3.98i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.98 + 1.14i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2.22T + 41T^{2} \) |
| 43 | \( 1 + 2.22iT - 43T^{2} \) |
| 47 | \( 1 + (-5.66 + 3.27i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.88 + 6.72i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.05 - 5.28i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.24 + 1.87i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.08 + 4.08i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 10.3iT - 71T^{2} \) |
| 73 | \( 1 + (-6.53 + 11.3i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.44 + 7.69i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 4.79iT - 83T^{2} \) |
| 89 | \( 1 + (-0.743 - 1.28i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 9.05T + 97T^{2} \) |
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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
−8.621393901625395743786080876279, −7.78937652391547221051937068493, −6.90766696723125998555899419850, −6.11020518636700744209865620998, −5.43544546831927095362299085367, −4.11908577311951512536680976573, −3.58788817656539528689838523492, −2.65964951326771132947509720813, −1.62563148943408353740758203468, −0.25902579465038281154963104098,
1.06949613114812199820613218275, 2.45688266444902407102400720523, 3.56382235870499748127590166850, 4.27306484259025456690861124352, 5.49436584334999311843495449342, 5.99054135644848483094994691910, 6.75187687975220582716539366730, 7.51219251409807124935967778334, 8.107490680736594390769117474892, 9.081373395235463327588507090907