| L(s) = 1 | + (−0.586 + 1.28i)2-s + (2.07 + 1.30i)3-s + (−1.31 − 1.50i)4-s + (−2.72 + 2.17i)5-s + (−2.90 + 1.90i)6-s + (−0.190 + 0.0435i)7-s + (2.71 − 0.803i)8-s + (1.31 + 2.73i)9-s + (−1.19 − 4.77i)10-s + (4.11 − 1.44i)11-s + (−0.756 − 4.85i)12-s + (0.215 − 0.447i)13-s + (0.0558 − 0.271i)14-s + (−8.49 + 0.957i)15-s + (−0.556 + 3.96i)16-s + (3.07 + 3.07i)17-s + ⋯ |
| L(s) = 1 | + (−0.414 + 0.909i)2-s + (1.20 + 0.754i)3-s + (−0.656 − 0.754i)4-s + (−1.21 + 0.970i)5-s + (−1.18 + 0.779i)6-s + (−0.0721 + 0.0164i)7-s + (0.958 − 0.284i)8-s + (0.438 + 0.910i)9-s + (−0.378 − 1.50i)10-s + (1.24 − 0.434i)11-s + (−0.218 − 1.40i)12-s + (0.0598 − 0.124i)13-s + (0.0149 − 0.0724i)14-s + (−2.19 + 0.247i)15-s + (−0.139 + 0.990i)16-s + (0.746 + 0.746i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.432 - 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.537065 + 0.852931i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.537065 + 0.852931i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.586 - 1.28i)T \) |
| 29 | \( 1 + (4.83 + 2.36i)T \) |
| good | 3 | \( 1 + (-2.07 - 1.30i)T + (1.30 + 2.70i)T^{2} \) |
| 5 | \( 1 + (2.72 - 2.17i)T + (1.11 - 4.87i)T^{2} \) |
| 7 | \( 1 + (0.190 - 0.0435i)T + (6.30 - 3.03i)T^{2} \) |
| 11 | \( 1 + (-4.11 + 1.44i)T + (8.60 - 6.85i)T^{2} \) |
| 13 | \( 1 + (-0.215 + 0.447i)T + (-8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (-3.07 - 3.07i)T + 17iT^{2} \) |
| 19 | \( 1 + (3.92 + 6.24i)T + (-8.24 + 17.1i)T^{2} \) |
| 23 | \( 1 + (-3.53 - 2.81i)T + (5.11 + 22.4i)T^{2} \) |
| 31 | \( 1 + (-5.78 - 0.652i)T + (30.2 + 6.89i)T^{2} \) |
| 37 | \( 1 + (-1.88 + 5.39i)T + (-28.9 - 23.0i)T^{2} \) |
| 41 | \( 1 + (4.93 - 4.93i)T - 41iT^{2} \) |
| 43 | \( 1 + (0.162 + 1.43i)T + (-41.9 + 9.56i)T^{2} \) |
| 47 | \( 1 + (1.36 + 3.90i)T + (-36.7 + 29.3i)T^{2} \) |
| 53 | \( 1 + (-0.304 - 0.382i)T + (-11.7 + 51.6i)T^{2} \) |
| 59 | \( 1 - 1.54iT - 59T^{2} \) |
| 61 | \( 1 + (1.69 - 2.70i)T + (-26.4 - 54.9i)T^{2} \) |
| 67 | \( 1 + (2.69 - 1.29i)T + (41.7 - 52.3i)T^{2} \) |
| 71 | \( 1 + (10.6 + 5.13i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (0.531 + 4.72i)T + (-71.1 + 16.2i)T^{2} \) |
| 79 | \( 1 + (-2.20 + 6.28i)T + (-61.7 - 49.2i)T^{2} \) |
| 83 | \( 1 + (10.5 + 2.40i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (0.947 - 8.41i)T + (-86.7 - 19.8i)T^{2} \) |
| 97 | \( 1 + (-4.31 - 6.86i)T + (-42.0 + 87.3i)T^{2} \) |
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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
−14.46328733303988826845087381538, −13.36036588100111594190665683902, −11.51538918527694017047076999847, −10.49462272963161720300273422961, −9.305720295505849286728878742834, −8.502035918860468971678192037532, −7.53791789009416119947833082438, −6.42472278911483691537433562058, −4.32174469209505253072037101880, −3.34908851750534410434047174143,
1.45407532295632168140838379044, 3.35043465004104678706560185206, 4.43735470162076939124536503556, 7.16997514001042420325537980900, 8.186339836966389953295852474368, 8.727338425077033287656266383359, 9.779063753395026165535176571162, 11.48844444558052604794757927195, 12.30728344507648644003435088953, 12.85925300861063470197568104346
