| L(s) = 1 | + (0.302 + 1.84i)2-s + (−0.267 − 0.963i)3-s + (−1.41 + 0.478i)4-s + (1.62 − 2.39i)5-s + (1.69 − 0.785i)6-s + (1.15 + 0.255i)7-s + (0.440 + 0.830i)8-s + (−0.856 + 0.515i)9-s + (4.90 + 2.26i)10-s + (3.14 + 2.39i)11-s + (0.840 + 1.23i)12-s + (−3.86 − 2.32i)13-s + (−0.120 + 2.21i)14-s + (−2.73 − 0.922i)15-s + (−3.78 + 2.87i)16-s + (0.853 − 0.187i)17-s + ⋯ |
| L(s) = 1 | + (0.213 + 1.30i)2-s + (−0.154 − 0.556i)3-s + (−0.709 + 0.239i)4-s + (0.724 − 1.06i)5-s + (0.692 − 0.320i)6-s + (0.438 + 0.0964i)7-s + (0.155 + 0.293i)8-s + (−0.285 + 0.171i)9-s + (1.55 + 0.717i)10-s + (0.948 + 0.721i)11-s + (0.242 + 0.357i)12-s + (−1.07 − 0.644i)13-s + (−0.0321 + 0.592i)14-s + (−0.706 − 0.238i)15-s + (−0.945 + 0.719i)16-s + (0.207 − 0.0455i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.30100 + 0.576309i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30100 + 0.576309i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (3.84 + 6.65i)T \) |
| good | 2 | \( 1 + (-0.302 - 1.84i)T + (-1.89 + 0.638i)T^{2} \) |
| 5 | \( 1 + (-1.62 + 2.39i)T + (-1.85 - 4.64i)T^{2} \) |
| 7 | \( 1 + (-1.15 - 0.255i)T + (6.35 + 2.93i)T^{2} \) |
| 11 | \( 1 + (-3.14 - 2.39i)T + (2.94 + 10.5i)T^{2} \) |
| 13 | \( 1 + (3.86 + 2.32i)T + (6.08 + 11.4i)T^{2} \) |
| 17 | \( 1 + (-0.853 + 0.187i)T + (15.4 - 7.13i)T^{2} \) |
| 19 | \( 1 + (3.75 + 0.408i)T + (18.5 + 4.08i)T^{2} \) |
| 23 | \( 1 + (-2.31 + 2.73i)T + (-3.72 - 22.6i)T^{2} \) |
| 29 | \( 1 + (0.671 - 4.09i)T + (-27.4 - 9.25i)T^{2} \) |
| 31 | \( 1 + (5.65 - 0.614i)T + (30.2 - 6.66i)T^{2} \) |
| 37 | \( 1 + (2.58 - 4.87i)T + (-20.7 - 30.6i)T^{2} \) |
| 41 | \( 1 + (4.30 + 5.06i)T + (-6.63 + 40.4i)T^{2} \) |
| 43 | \( 1 + (-7.31 + 5.55i)T + (11.5 - 41.4i)T^{2} \) |
| 47 | \( 1 + (-5.31 - 7.83i)T + (-17.3 + 43.6i)T^{2} \) |
| 53 | \( 1 + (9.18 - 4.25i)T + (34.3 - 40.3i)T^{2} \) |
| 61 | \( 1 + (1.88 + 11.5i)T + (-57.8 + 19.4i)T^{2} \) |
| 67 | \( 1 + (0.875 + 1.65i)T + (-37.5 + 55.4i)T^{2} \) |
| 71 | \( 1 + (-2.55 - 3.76i)T + (-26.2 + 65.9i)T^{2} \) |
| 73 | \( 1 + (0.451 - 8.31i)T + (-72.5 - 7.89i)T^{2} \) |
| 79 | \( 1 + (3.71 - 13.3i)T + (-67.6 - 40.7i)T^{2} \) |
| 83 | \( 1 + (-11.1 + 10.5i)T + (4.49 - 82.8i)T^{2} \) |
| 89 | \( 1 + (0.307 - 1.87i)T + (-84.3 - 28.4i)T^{2} \) |
| 97 | \( 1 + (-0.436 - 8.04i)T + (-96.4 + 10.4i)T^{2} \) |
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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
−12.78413114556473454763545741619, −12.37019075529938483711994977873, −10.88226823691577074881716942585, −9.398117445285057895878181530093, −8.541824284176575779239807678071, −7.46710008545126669156461849395, −6.54759986905108544138475715031, −5.38991135526924803802629407265, −4.70992013647189329525229190203, −1.85323001669890736194250470968,
2.00927962582800908760034307963, 3.29810081308231731247320372670, 4.48154594994766287306576874694, 6.07281896777399558816376800853, 7.22684370243229054463140763325, 9.110309177724812290454300028942, 9.881526100752583915822194189943, 10.79051521699791827671419102202, 11.34657467175182376544302560460, 12.23901653960298159516816423934
