| L(s) = 1 | + (−2.17 + 0.479i)2-s + (−1.34 + 1.08i)3-s + (2.69 − 1.24i)4-s + (2.10 + 0.583i)5-s + (2.41 − 3.01i)6-s + (0.124 − 0.117i)7-s + (−1.72 + 1.31i)8-s + (0.634 − 2.93i)9-s + (−4.85 − 0.263i)10-s + (3.16 + 3.72i)11-s + (−2.28 + 4.61i)12-s + (−0.472 − 1.40i)13-s + (−0.214 + 0.316i)14-s + (−3.46 + 1.49i)15-s + (−0.713 + 0.839i)16-s + (−4.34 + 4.58i)17-s + ⋯ |
| L(s) = 1 | + (−1.54 + 0.339i)2-s + (−0.778 + 0.627i)3-s + (1.34 − 0.624i)4-s + (0.940 + 0.261i)5-s + (0.985 − 1.23i)6-s + (0.0470 − 0.0445i)7-s + (−0.611 + 0.464i)8-s + (0.211 − 0.977i)9-s + (−1.53 − 0.0833i)10-s + (0.953 + 1.12i)11-s + (−0.658 + 1.33i)12-s + (−0.131 − 0.388i)13-s + (−0.0573 + 0.0846i)14-s + (−0.895 + 0.387i)15-s + (−0.178 + 0.209i)16-s + (−1.05 + 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.179 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.179 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.319635 + 0.383350i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.319635 + 0.383350i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.34 - 1.08i)T \) |
| 59 | \( 1 + (2.05 + 7.40i)T \) |
| good | 2 | \( 1 + (2.17 - 0.479i)T + (1.81 - 0.839i)T^{2} \) |
| 5 | \( 1 + (-2.10 - 0.583i)T + (4.28 + 2.57i)T^{2} \) |
| 7 | \( 1 + (-0.124 + 0.117i)T + (0.378 - 6.98i)T^{2} \) |
| 11 | \( 1 + (-3.16 - 3.72i)T + (-1.77 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.472 + 1.40i)T + (-10.3 + 7.86i)T^{2} \) |
| 17 | \( 1 + (4.34 - 4.58i)T + (-0.920 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-2.44 - 6.14i)T + (-13.7 + 13.0i)T^{2} \) |
| 23 | \( 1 + (1.31 - 0.142i)T + (22.4 - 4.94i)T^{2} \) |
| 29 | \( 1 + (-1.17 + 5.34i)T + (-26.3 - 12.1i)T^{2} \) |
| 31 | \( 1 + (-4.65 - 1.85i)T + (22.5 + 21.3i)T^{2} \) |
| 37 | \( 1 + (0.190 - 0.251i)T + (-9.89 - 35.6i)T^{2} \) |
| 41 | \( 1 + (0.377 - 3.47i)T + (-40.0 - 8.81i)T^{2} \) |
| 43 | \( 1 + (0.826 + 0.702i)T + (6.95 + 42.4i)T^{2} \) |
| 47 | \( 1 + (-1.77 - 6.39i)T + (-40.2 + 24.2i)T^{2} \) |
| 53 | \( 1 + (-4.06 + 0.220i)T + (52.6 - 5.73i)T^{2} \) |
| 61 | \( 1 + (2.93 + 13.3i)T + (-55.3 + 25.6i)T^{2} \) |
| 67 | \( 1 + (-6.87 - 9.03i)T + (-17.9 + 64.5i)T^{2} \) |
| 71 | \( 1 + (4.68 - 1.30i)T + (60.8 - 36.6i)T^{2} \) |
| 73 | \( 1 + (-6.10 - 4.14i)T + (27.0 + 67.8i)T^{2} \) |
| 79 | \( 1 + (1.68 + 10.2i)T + (-74.8 + 25.2i)T^{2} \) |
| 83 | \( 1 + (1.50 + 2.83i)T + (-46.5 + 68.6i)T^{2} \) |
| 89 | \( 1 + (-6.77 - 1.49i)T + (80.7 + 37.3i)T^{2} \) |
| 97 | \( 1 + (-6.32 + 4.28i)T + (35.9 - 90.1i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.68884413779525108387021285552, −11.59720016187023697750315681791, −10.44667819763816220364157977823, −9.937983623840410278445176414645, −9.291640267196456400500130773454, −8.002964686558882500564375682442, −6.63106694137469471803918042098, −6.02335924973459518910369325380, −4.27958964190155288706813331597, −1.68930429492624472671123527466,
0.889557902884467592750397072812, 2.33131837662806438452438814486, 5.06974582140002012294634982636, 6.44598770831050617895974197886, 7.23454802695802025504112913899, 8.724304435263846616725826751156, 9.257526934781630048824141172854, 10.39979379073824300719999027302, 11.40495347162502039772568434309, 11.82204730633683511634598694717
