| L(s) = 1 | + (−0.651 − 2.34i)2-s + (−0.723 + 1.57i)3-s + (−3.36 + 2.02i)4-s + (−1.17 + 0.0639i)5-s + (4.16 + 0.673i)6-s + (0.423 + 2.58i)7-s + (3.39 + 3.21i)8-s + (−1.95 − 2.27i)9-s + (0.917 + 2.72i)10-s + (2.48 + 4.68i)11-s + (−0.749 − 6.75i)12-s + (3.96 + 1.58i)13-s + (5.77 − 2.67i)14-s + (0.753 − 1.90i)15-s + (1.66 − 3.13i)16-s + (−3.67 − 0.602i)17-s + ⋯ |
| L(s) = 1 | + (−0.460 − 1.65i)2-s + (−0.417 + 0.908i)3-s + (−1.68 + 1.01i)4-s + (−0.527 + 0.0285i)5-s + (1.69 + 0.274i)6-s + (0.160 + 0.976i)7-s + (1.20 + 1.13i)8-s + (−0.650 − 0.759i)9-s + (0.290 + 0.861i)10-s + (0.748 + 1.41i)11-s + (−0.216 − 1.94i)12-s + (1.10 + 0.438i)13-s + (1.54 − 0.714i)14-s + (0.194 − 0.491i)15-s + (0.415 − 0.782i)16-s + (−0.890 − 0.146i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.446i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.548358 + 0.129114i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.548358 + 0.129114i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.723 - 1.57i)T \) |
| 59 | \( 1 + (1.60 - 7.51i)T \) |
| good | 2 | \( 1 + (0.651 + 2.34i)T + (-1.71 + 1.03i)T^{2} \) |
| 5 | \( 1 + (1.17 - 0.0639i)T + (4.97 - 0.540i)T^{2} \) |
| 7 | \( 1 + (-0.423 - 2.58i)T + (-6.63 + 2.23i)T^{2} \) |
| 11 | \( 1 + (-2.48 - 4.68i)T + (-6.17 + 9.10i)T^{2} \) |
| 13 | \( 1 + (-3.96 - 1.58i)T + (9.43 + 8.94i)T^{2} \) |
| 17 | \( 1 + (3.67 + 0.602i)T + (16.1 + 5.42i)T^{2} \) |
| 19 | \( 1 + (3.11 - 3.67i)T + (-3.07 - 18.7i)T^{2} \) |
| 23 | \( 1 + (0.491 + 0.373i)T + (6.15 + 22.1i)T^{2} \) |
| 29 | \( 1 + (6.72 + 1.86i)T + (24.8 + 14.9i)T^{2} \) |
| 31 | \( 1 + (-3.05 + 2.59i)T + (5.01 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-4.02 - 4.25i)T + (-2.00 + 36.9i)T^{2} \) |
| 41 | \( 1 + (-4.87 - 6.41i)T + (-10.9 + 39.5i)T^{2} \) |
| 43 | \( 1 + (3.31 + 1.75i)T + (24.1 + 35.5i)T^{2} \) |
| 47 | \( 1 + (0.0491 - 0.907i)T + (-46.7 - 5.08i)T^{2} \) |
| 53 | \( 1 + (-2.05 + 6.10i)T + (-42.1 - 32.0i)T^{2} \) |
| 61 | \( 1 + (-7.83 + 2.17i)T + (52.2 - 31.4i)T^{2} \) |
| 67 | \( 1 + (-8.13 + 8.59i)T + (-3.62 - 66.9i)T^{2} \) |
| 71 | \( 1 + (7.99 + 0.433i)T + (70.5 + 7.67i)T^{2} \) |
| 73 | \( 1 + (4.35 + 9.41i)T + (-47.2 + 55.6i)T^{2} \) |
| 79 | \( 1 + (-2.52 - 3.71i)T + (-29.2 + 73.3i)T^{2} \) |
| 83 | \( 1 + (-13.8 + 3.04i)T + (75.3 - 34.8i)T^{2} \) |
| 89 | \( 1 + (3.44 - 12.4i)T + (-76.2 - 45.8i)T^{2} \) |
| 97 | \( 1 + (-2.47 + 5.34i)T + (-62.7 - 73.9i)T^{2} \) |
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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
−12.15074537982033424852801973875, −11.65952135271712077377629028053, −10.98048373461051409724159342379, −9.823821968103066260124791352010, −9.214647306866931885679738313556, −8.291147535533656651102648584285, −6.23704292019823459796827988678, −4.48224822416938287969521585061, −3.78304212232050929165989758986, −2.05375300637541254014640565941,
0.65641771885128677480878008313, 4.06131864181192313632934867333, 5.68980787835171657943339631600, 6.48918687173640661776683797735, 7.34016770008223974758300327482, 8.281374575647875239405411508546, 8.917640644655947301342939869210, 10.79739884065785768703414440606, 11.42165016542829661818798036413, 13.14137734724241657446623436969
