| L(s) = 1 | + (0.277 − 0.166i)2-s + (1.51 + 0.836i)3-s + (−0.887 + 1.67i)4-s + (0.0154 + 0.141i)5-s + (0.559 − 0.0212i)6-s + (−0.858 + 0.289i)7-s + (0.0682 + 1.25i)8-s + (1.60 + 2.53i)9-s + (0.0279 + 0.0367i)10-s + (1.74 − 2.57i)11-s + (−2.74 + 1.79i)12-s + (−0.152 + 0.161i)13-s + (−0.189 + 0.223i)14-s + (−0.0952 + 0.228i)15-s + (−1.89 − 2.80i)16-s + (0.541 − 1.60i)17-s + ⋯ |
| L(s) = 1 | + (0.195 − 0.117i)2-s + (0.875 + 0.482i)3-s + (−0.443 + 0.837i)4-s + (0.00690 + 0.0634i)5-s + (0.228 − 0.00865i)6-s + (−0.324 + 0.109i)7-s + (0.0241 + 0.444i)8-s + (0.533 + 0.845i)9-s + (0.00883 + 0.0116i)10-s + (0.526 − 0.775i)11-s + (−0.792 + 0.518i)12-s + (−0.0423 + 0.0447i)13-s + (−0.0506 + 0.0596i)14-s + (−0.0245 + 0.0588i)15-s + (−0.474 − 0.700i)16-s + (0.131 − 0.389i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.34273 + 0.636767i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.34273 + 0.636767i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.51 - 0.836i)T \) |
| 59 | \( 1 + (-3.93 + 6.59i)T \) |
| good | 2 | \( 1 + (-0.277 + 0.166i)T + (0.936 - 1.76i)T^{2} \) |
| 5 | \( 1 + (-0.0154 - 0.141i)T + (-4.88 + 1.07i)T^{2} \) |
| 7 | \( 1 + (0.858 - 0.289i)T + (5.57 - 4.23i)T^{2} \) |
| 11 | \( 1 + (-1.74 + 2.57i)T + (-4.07 - 10.2i)T^{2} \) |
| 13 | \( 1 + (0.152 - 0.161i)T + (-0.703 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.541 + 1.60i)T + (-13.5 - 10.2i)T^{2} \) |
| 19 | \( 1 + (-0.332 - 2.02i)T + (-18.0 + 6.06i)T^{2} \) |
| 23 | \( 1 + (2.08 + 7.49i)T + (-19.7 + 11.8i)T^{2} \) |
| 29 | \( 1 + (-1.29 + 2.15i)T + (-13.5 - 25.6i)T^{2} \) |
| 31 | \( 1 + (-3.42 - 0.561i)T + (29.3 + 9.89i)T^{2} \) |
| 37 | \( 1 + (4.70 + 0.254i)T + (36.7 + 4.00i)T^{2} \) |
| 41 | \( 1 + (7.40 + 2.05i)T + (35.1 + 21.1i)T^{2} \) |
| 43 | \( 1 + (-1.53 + 1.04i)T + (15.9 - 39.9i)T^{2} \) |
| 47 | \( 1 + (1.35 + 0.147i)T + (45.9 + 10.1i)T^{2} \) |
| 53 | \( 1 + (6.14 - 8.08i)T + (-14.1 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.68 + 6.12i)T + (-28.5 + 53.8i)T^{2} \) |
| 67 | \( 1 + (-4.55 + 0.246i)T + (66.6 - 7.24i)T^{2} \) |
| 71 | \( 1 + (-0.00120 + 0.0110i)T + (-69.3 - 15.2i)T^{2} \) |
| 73 | \( 1 + (-6.41 - 5.45i)T + (11.8 + 72.0i)T^{2} \) |
| 79 | \( 1 + (4.43 - 11.1i)T + (-57.3 - 54.3i)T^{2} \) |
| 83 | \( 1 + (8.16 - 3.77i)T + (53.7 - 63.2i)T^{2} \) |
| 89 | \( 1 + (-15.2 - 9.18i)T + (41.6 + 78.6i)T^{2} \) |
| 97 | \( 1 + (2.61 - 2.22i)T + (15.6 - 95.7i)T^{2} \) |
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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
−12.87343401135296278717571552705, −12.06867027550682955087978950361, −10.79334842490063823638334817390, −9.660495697178732402355289380701, −8.685501942663990611099576300415, −8.074858192775164889429921159136, −6.65745650958196068585483997202, −4.87894289520460951078283771180, −3.74472838081817662404305029612, −2.73676021393809051565322037863,
1.57067455810030818160240308586, 3.50996576189877042017675945829, 4.86327340597815663143935495091, 6.35361691958115734419983590508, 7.25356764904309704712956278827, 8.624229082257822258241561260088, 9.514491156680361578537994707612, 10.22954152684157253332416463792, 11.77261806372835425318632062751, 12.89295677142253444166775418430
