| L(s) = 1 | + (0.205 + 1.25i)2-s + (0.267 + 0.963i)3-s + (0.372 − 0.125i)4-s + (1.50 − 2.21i)5-s + (−1.15 + 0.532i)6-s + (1.03 + 0.226i)7-s + (1.42 + 2.68i)8-s + (−0.856 + 0.515i)9-s + (3.08 + 1.42i)10-s + (−3.22 − 2.45i)11-s + (0.220 + 0.325i)12-s + (−0.227 − 0.136i)13-s + (−0.0724 + 1.33i)14-s + (2.54 + 0.855i)15-s + (−2.43 + 1.85i)16-s + (−6.62 + 1.45i)17-s + ⋯ |
| L(s) = 1 | + (0.145 + 0.884i)2-s + (0.154 + 0.556i)3-s + (0.186 − 0.0627i)4-s + (0.672 − 0.992i)5-s + (−0.469 + 0.217i)6-s + (0.389 + 0.0857i)7-s + (0.502 + 0.947i)8-s + (−0.285 + 0.171i)9-s + (0.975 + 0.451i)10-s + (−0.973 − 0.740i)11-s + (0.0636 + 0.0939i)12-s + (−0.0630 − 0.0379i)13-s + (−0.0193 + 0.357i)14-s + (0.655 + 0.220i)15-s + (−0.608 + 0.462i)16-s + (−1.60 + 0.353i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.30321 + 0.815426i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30321 + 0.815426i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.267 - 0.963i)T \) |
| 59 | \( 1 + (-5.55 - 5.30i)T \) |
| good | 2 | \( 1 + (-0.205 - 1.25i)T + (-1.89 + 0.638i)T^{2} \) |
| 5 | \( 1 + (-1.50 + 2.21i)T + (-1.85 - 4.64i)T^{2} \) |
| 7 | \( 1 + (-1.03 - 0.226i)T + (6.35 + 2.93i)T^{2} \) |
| 11 | \( 1 + (3.22 + 2.45i)T + (2.94 + 10.5i)T^{2} \) |
| 13 | \( 1 + (0.227 + 0.136i)T + (6.08 + 11.4i)T^{2} \) |
| 17 | \( 1 + (6.62 - 1.45i)T + (15.4 - 7.13i)T^{2} \) |
| 19 | \( 1 + (0.399 + 0.0434i)T + (18.5 + 4.08i)T^{2} \) |
| 23 | \( 1 + (4.63 - 5.46i)T + (-3.72 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.42 + 8.68i)T + (-27.4 - 9.25i)T^{2} \) |
| 31 | \( 1 + (-9.46 + 1.02i)T + (30.2 - 6.66i)T^{2} \) |
| 37 | \( 1 + (-0.879 + 1.65i)T + (-20.7 - 30.6i)T^{2} \) |
| 41 | \( 1 + (2.64 + 3.11i)T + (-6.63 + 40.4i)T^{2} \) |
| 43 | \( 1 + (3.42 - 2.60i)T + (11.5 - 41.4i)T^{2} \) |
| 47 | \( 1 + (-3.05 - 4.50i)T + (-17.3 + 43.6i)T^{2} \) |
| 53 | \( 1 + (-3.99 + 1.84i)T + (34.3 - 40.3i)T^{2} \) |
| 61 | \( 1 + (0.729 + 4.44i)T + (-57.8 + 19.4i)T^{2} \) |
| 67 | \( 1 + (-1.35 - 2.55i)T + (-37.5 + 55.4i)T^{2} \) |
| 71 | \( 1 + (2.05 + 3.03i)T + (-26.2 + 65.9i)T^{2} \) |
| 73 | \( 1 + (-0.589 + 10.8i)T + (-72.5 - 7.89i)T^{2} \) |
| 79 | \( 1 + (2.40 - 8.67i)T + (-67.6 - 40.7i)T^{2} \) |
| 83 | \( 1 + (-6.58 + 6.24i)T + (4.49 - 82.8i)T^{2} \) |
| 89 | \( 1 + (1.87 - 11.4i)T + (-84.3 - 28.4i)T^{2} \) |
| 97 | \( 1 + (-0.678 - 12.5i)T + (-96.4 + 10.4i)T^{2} \) |
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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
−13.52077860201159228469175046902, −11.78729189930222033271310235859, −10.84690933343410951795976347689, −9.762657392905472797859810123735, −8.544736537286172186297812226613, −7.947575873653147931476337331458, −6.28143243173704339177583594341, −5.44167354423400043704020541378, −4.50201214022855717583671280564, −2.23969527460207676782310257593,
2.07166991632264975424503647789, 2.80280949586246623819429553374, 4.62828874643407477712464050876, 6.48764909260788738290043040658, 7.09257534300239535187266966977, 8.420136823305305079423431219243, 10.04050941947289147087817581522, 10.55657042916826516059840665009, 11.50701289332021791203891779455, 12.49299388195444467265128861356
