L(s) = 1 | + (0.809 − 0.587i)2-s + (−0.785 − 2.41i)3-s + (0.309 − 0.951i)4-s + (−0.809 − 0.587i)5-s + (−2.05 − 1.49i)6-s + (−0.309 + 0.951i)7-s + (−0.309 − 0.951i)8-s + (−2.80 + 2.03i)9-s − 10-s + (−2.31 + 2.37i)11-s − 2.54·12-s + (−5.19 + 3.77i)13-s + (0.309 + 0.951i)14-s + (−0.785 + 2.41i)15-s + (−0.809 − 0.587i)16-s + (−3.19 − 2.32i)17-s + ⋯ |
L(s) = 1 | + (0.572 − 0.415i)2-s + (−0.453 − 1.39i)3-s + (0.154 − 0.475i)4-s + (−0.361 − 0.262i)5-s + (−0.840 − 0.610i)6-s + (−0.116 + 0.359i)7-s + (−0.109 − 0.336i)8-s + (−0.935 + 0.679i)9-s − 0.316·10-s + (−0.698 + 0.716i)11-s − 0.734·12-s + (−1.44 + 1.04i)13-s + (0.0825 + 0.254i)14-s + (−0.202 + 0.624i)15-s + (−0.202 − 0.146i)16-s + (−0.775 − 0.563i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.257966 + 0.415815i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.257966 + 0.415815i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 5 | \( 1 + (0.809 + 0.587i)T \) |
| 7 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 + (2.31 - 2.37i)T \) |
good | 3 | \( 1 + (0.785 + 2.41i)T + (-2.42 + 1.76i)T^{2} \) |
| 13 | \( 1 + (5.19 - 3.77i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (3.19 + 2.32i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (0.762 + 2.34i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 4.99T + 23T^{2} \) |
| 29 | \( 1 + (-2.79 + 8.60i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (6.38 - 4.63i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-3.41 + 10.5i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.14 - 6.61i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 3.02T + 43T^{2} \) |
| 47 | \( 1 + (2.58 + 7.96i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (8.14 - 5.91i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (1.26 - 3.89i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (2.91 + 2.11i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 1.28T + 67T^{2} \) |
| 71 | \( 1 + (-4.77 - 3.46i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-3.16 + 9.75i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (5.02 - 3.65i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (10.7 + 7.84i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + (-9.67 + 7.03i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.743216692410269824890921980188, −8.964693466377806670845382582403, −7.58543691733275132690647396414, −7.13501654448164634095245117916, −6.30602019735782191135077472362, −5.11641385212134864879735104006, −4.47571136280249407088756575802, −2.65225783730698662989270318045, −1.93548848432849530148972990373, −0.19984033280902606974979048565,
2.88921773760290214305140748213, 3.66972018302296327565319215062, 4.79665329787478811301377527847, 5.23446651835395171791942252161, 6.31995536976242121955918335639, 7.41013939258990265513410581571, 8.241076554123841442915354532556, 9.322158110860574120315218629178, 10.30413806261995455622121465960, 10.77194143710165034865979618569