| L(s) = 1 | + (1.16 + 0.358i)2-s + (0.190 − 0.484i)3-s + (−0.429 − 0.292i)4-s + (−2.56 + 0.386i)5-s + (0.394 − 0.495i)6-s + (−0.339 + 2.62i)7-s + (−1.91 − 2.39i)8-s + (2.00 + 1.85i)9-s + (−3.12 − 0.470i)10-s + (2.81 − 2.61i)11-s + (−0.223 + 0.152i)12-s + (−0.203 − 0.891i)13-s + (−1.33 + 2.92i)14-s + (−0.300 + 1.31i)15-s + (−0.982 − 2.50i)16-s + (−0.368 − 4.91i)17-s + ⋯ |
| L(s) = 1 | + (0.822 + 0.253i)2-s + (0.109 − 0.279i)3-s + (−0.214 − 0.146i)4-s + (−1.14 + 0.172i)5-s + (0.161 − 0.202i)6-s + (−0.128 + 0.991i)7-s + (−0.675 − 0.847i)8-s + (0.666 + 0.618i)9-s + (−0.987 − 0.148i)10-s + (0.849 − 0.788i)11-s + (−0.0645 + 0.0440i)12-s + (−0.0564 − 0.247i)13-s + (−0.356 + 0.782i)14-s + (−0.0776 + 0.340i)15-s + (−0.245 − 0.626i)16-s + (−0.0894 − 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.999176 + 0.0422449i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.999176 + 0.0422449i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (0.339 - 2.62i)T \) |
| good | 2 | \( 1 + (-1.16 - 0.358i)T + (1.65 + 1.12i)T^{2} \) |
| 3 | \( 1 + (-0.190 + 0.484i)T + (-2.19 - 2.04i)T^{2} \) |
| 5 | \( 1 + (2.56 - 0.386i)T + (4.77 - 1.47i)T^{2} \) |
| 11 | \( 1 + (-2.81 + 2.61i)T + (0.822 - 10.9i)T^{2} \) |
| 13 | \( 1 + (0.203 + 0.891i)T + (-11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (0.368 + 4.91i)T + (-16.8 + 2.53i)T^{2} \) |
| 19 | \( 1 + (1.24 - 2.15i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.295 - 3.93i)T + (-22.7 - 3.42i)T^{2} \) |
| 29 | \( 1 + (4.93 + 2.37i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (-4.93 - 8.54i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.81 + 1.23i)T + (13.5 - 34.4i)T^{2} \) |
| 41 | \( 1 + (1.31 + 1.65i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (2.67 - 3.35i)T + (-9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (5.46 + 1.68i)T + (38.8 + 26.4i)T^{2} \) |
| 53 | \( 1 + (6.34 + 4.32i)T + (19.3 + 49.3i)T^{2} \) |
| 59 | \( 1 + (-9.59 - 1.44i)T + (56.3 + 17.3i)T^{2} \) |
| 61 | \( 1 + (-11.9 + 8.14i)T + (22.2 - 56.7i)T^{2} \) |
| 67 | \( 1 + (3.38 + 5.85i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.32 + 2.56i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-10.8 + 3.35i)T + (60.3 - 41.1i)T^{2} \) |
| 79 | \( 1 + (-2.95 + 5.12i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.604 - 2.64i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (0.713 + 0.661i)T + (6.65 + 88.7i)T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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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
−15.56662196553261089524868802401, −14.52197767321921603737000908010, −13.47158532507654237521572577264, −12.32938181288658535509815574196, −11.43115244079773607413552257782, −9.562095201177600463837260883522, −8.171636040829489726454549466080, −6.67180584137192149555798032824, −5.11646778017490077665791833755, −3.55226075097595494194110660198,
3.93194712304552682579414452600, 4.30004690473743238141842131938, 6.77852014847892746419025111133, 8.254526181986029254272932210187, 9.712799543875635767407804054332, 11.29447715005069186015424899594, 12.35433388069749831136553540796, 13.16170726478226792128565812534, 14.62380521748912907745594891406, 15.23314293327015122792491809098
