| L(s) = 1 | + (1.32 + 0.445i)2-s + (0.856 + 0.515i)3-s + (−0.0401 − 0.0305i)4-s + (−1.32 + 3.31i)5-s + (0.903 + 1.06i)6-s + (3.77 − 1.74i)7-s + (−1.60 − 2.36i)8-s + (0.468 + 0.883i)9-s + (−3.22 + 3.79i)10-s + (−0.526 + 1.89i)11-s + (−0.0186 − 0.0468i)12-s + (0.345 − 0.652i)13-s + (5.76 − 0.627i)14-s + (−2.84 + 2.15i)15-s + (−1.04 − 3.75i)16-s + (−6.27 − 2.90i)17-s + ⋯ |
| L(s) = 1 | + (0.935 + 0.315i)2-s + (0.494 + 0.297i)3-s + (−0.0200 − 0.0152i)4-s + (−0.590 + 1.48i)5-s + (0.369 + 0.434i)6-s + (1.42 − 0.659i)7-s + (−0.568 − 0.837i)8-s + (0.156 + 0.294i)9-s + (−1.02 + 1.20i)10-s + (−0.158 + 0.571i)11-s + (−0.00538 − 0.0135i)12-s + (0.0958 − 0.180i)13-s + (1.54 − 0.167i)14-s + (−0.733 + 0.557i)15-s + (−0.260 − 0.938i)16-s + (−1.52 − 0.703i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.71676 + 0.769017i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.71676 + 0.769017i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.856 - 0.515i)T \) |
| 59 | \( 1 + (-4.07 - 6.51i)T \) |
| good | 2 | \( 1 + (-1.32 - 0.445i)T + (1.59 + 1.21i)T^{2} \) |
| 5 | \( 1 + (1.32 - 3.31i)T + (-3.62 - 3.43i)T^{2} \) |
| 7 | \( 1 + (-3.77 + 1.74i)T + (4.53 - 5.33i)T^{2} \) |
| 11 | \( 1 + (0.526 - 1.89i)T + (-9.42 - 5.67i)T^{2} \) |
| 13 | \( 1 + (-0.345 + 0.652i)T + (-7.29 - 10.7i)T^{2} \) |
| 17 | \( 1 + (6.27 + 2.90i)T + (11.0 + 12.9i)T^{2} \) |
| 19 | \( 1 + (-2.03 + 0.447i)T + (17.2 - 7.97i)T^{2} \) |
| 23 | \( 1 + (-1.23 + 7.53i)T + (-21.7 - 7.34i)T^{2} \) |
| 29 | \( 1 + (-0.855 + 0.288i)T + (23.0 - 17.5i)T^{2} \) |
| 31 | \( 1 + (0.00573 + 0.00126i)T + (28.1 + 13.0i)T^{2} \) |
| 37 | \( 1 + (4.72 - 6.97i)T + (-13.6 - 34.3i)T^{2} \) |
| 41 | \( 1 + (-0.866 - 5.28i)T + (-38.8 + 13.0i)T^{2} \) |
| 43 | \( 1 + (1.53 + 5.52i)T + (-36.8 + 22.1i)T^{2} \) |
| 47 | \( 1 + (-3.13 - 7.86i)T + (-34.1 + 32.3i)T^{2} \) |
| 53 | \( 1 + (3.07 + 3.61i)T + (-8.57 + 52.3i)T^{2} \) |
| 61 | \( 1 + (-4.95 - 1.67i)T + (48.5 + 36.9i)T^{2} \) |
| 67 | \( 1 + (-4.50 - 6.63i)T + (-24.7 + 62.2i)T^{2} \) |
| 71 | \( 1 + (2.77 + 6.96i)T + (-51.5 + 48.8i)T^{2} \) |
| 73 | \( 1 + (7.48 - 0.813i)T + (71.2 - 15.6i)T^{2} \) |
| 79 | \( 1 + (14.4 - 8.68i)T + (37.0 - 69.7i)T^{2} \) |
| 83 | \( 1 + (-0.133 - 2.45i)T + (-82.5 + 8.97i)T^{2} \) |
| 89 | \( 1 + (-14.6 + 4.93i)T + (70.8 - 53.8i)T^{2} \) |
| 97 | \( 1 + (-5.23 - 0.569i)T + (94.7 + 20.8i)T^{2} \) |
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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
−13.23233665764464528588468635335, −11.76734829573127158763618241065, −10.90362560486949448273254285844, −10.10599634045751924177292819598, −8.592476931911023013102159977067, −7.37414120649165853435909197984, −6.66687685615506196627088207342, −4.85592198877041908873748153186, −4.16953213635087629115698617308, −2.75231412862487427533595556683,
1.88606499885099807050198673951, 3.78168973015738493039533338988, 4.79736881770038721879519202217, 5.59782853428001582416448080501, 7.73712982547465659328920936814, 8.666162015388796642218290761361, 8.923465131452093494672951778513, 11.23348968731804681405463995604, 11.76937217081823231806321704459, 12.67855726109387010214891172336
