| L(s) = 1 | + (0.239 + 0.862i)2-s + (−1.56 − 0.749i)3-s + (1.02 − 0.618i)4-s + (0.401 − 0.0217i)5-s + (0.272 − 1.52i)6-s + (−0.709 − 4.32i)7-s + (2.07 + 1.96i)8-s + (1.87 + 2.34i)9-s + (0.114 + 0.340i)10-s + (−1.91 − 3.61i)11-s + (−2.06 + 0.195i)12-s + (4.37 + 1.74i)13-s + (3.56 − 1.64i)14-s + (−0.642 − 0.266i)15-s + (−0.0767 + 0.144i)16-s + (3.68 + 0.604i)17-s + ⋯ |
| L(s) = 1 | + (0.169 + 0.609i)2-s + (−0.901 − 0.432i)3-s + (0.513 − 0.309i)4-s + (0.179 − 0.00972i)5-s + (0.111 − 0.622i)6-s + (−0.268 − 1.63i)7-s + (0.734 + 0.696i)8-s + (0.625 + 0.780i)9-s + (0.0363 + 0.107i)10-s + (−0.577 − 1.08i)11-s + (−0.596 + 0.0564i)12-s + (1.21 + 0.483i)13-s + (0.951 − 0.440i)14-s + (−0.165 − 0.0688i)15-s + (−0.0191 + 0.0362i)16-s + (0.893 + 0.146i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.12245 - 0.237776i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12245 - 0.237776i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.56 + 0.749i)T \) |
| 59 | \( 1 + (-3.65 - 6.75i)T \) |
| good | 2 | \( 1 + (-0.239 - 0.862i)T + (-1.71 + 1.03i)T^{2} \) |
| 5 | \( 1 + (-0.401 + 0.0217i)T + (4.97 - 0.540i)T^{2} \) |
| 7 | \( 1 + (0.709 + 4.32i)T + (-6.63 + 2.23i)T^{2} \) |
| 11 | \( 1 + (1.91 + 3.61i)T + (-6.17 + 9.10i)T^{2} \) |
| 13 | \( 1 + (-4.37 - 1.74i)T + (9.43 + 8.94i)T^{2} \) |
| 17 | \( 1 + (-3.68 - 0.604i)T + (16.1 + 5.42i)T^{2} \) |
| 19 | \( 1 + (1.96 - 2.30i)T + (-3.07 - 18.7i)T^{2} \) |
| 23 | \( 1 + (-1.95 - 1.48i)T + (6.15 + 22.1i)T^{2} \) |
| 29 | \( 1 + (6.32 + 1.75i)T + (24.8 + 14.9i)T^{2} \) |
| 31 | \( 1 + (4.82 - 4.10i)T + (5.01 - 30.5i)T^{2} \) |
| 37 | \( 1 + (0.558 + 0.589i)T + (-2.00 + 36.9i)T^{2} \) |
| 41 | \( 1 + (-1.47 - 1.94i)T + (-10.9 + 39.5i)T^{2} \) |
| 43 | \( 1 + (-5.47 - 2.90i)T + (24.1 + 35.5i)T^{2} \) |
| 47 | \( 1 + (0.623 - 11.4i)T + (-46.7 - 5.08i)T^{2} \) |
| 53 | \( 1 + (-3.45 + 10.2i)T + (-42.1 - 32.0i)T^{2} \) |
| 61 | \( 1 + (0.368 - 0.102i)T + (52.2 - 31.4i)T^{2} \) |
| 67 | \( 1 + (-1.33 + 1.40i)T + (-3.62 - 66.9i)T^{2} \) |
| 71 | \( 1 + (-3.68 - 0.199i)T + (70.5 + 7.67i)T^{2} \) |
| 73 | \( 1 + (-2.95 - 6.38i)T + (-47.2 + 55.6i)T^{2} \) |
| 79 | \( 1 + (-3.84 - 5.67i)T + (-29.2 + 73.3i)T^{2} \) |
| 83 | \( 1 + (6.59 - 1.45i)T + (75.3 - 34.8i)T^{2} \) |
| 89 | \( 1 + (2.26 - 8.14i)T + (-76.2 - 45.8i)T^{2} \) |
| 97 | \( 1 + (-0.983 + 2.12i)T + (-62.7 - 73.9i)T^{2} \) |
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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
−12.87801042344759978247509941271, −11.20821286635699516112135319289, −10.96969242256478979346016389435, −9.986272212056157057080999065840, −8.023174832098555799031426006669, −7.24909773792874379585357147396, −6.22571599418617122109213583598, −5.52964012298059272299520515420, −3.87764615166735490434589585398, −1.30593881502552781397598095359,
2.17231154602315293488634709075, 3.68058613873906179098940358446, 5.29715988531736234056801074967, 6.14439237225620682406735053838, 7.47031026309659022742143161416, 9.013205419992991299661649344859, 10.04141812568555860688323135796, 10.95167277112273161352826697249, 11.79723580383186746769639620308, 12.55420597076945131563214421698
