| L(s) = 1 | + (−1.45 − 0.581i)2-s + (1.72 + 0.187i)3-s + (−1.11 − 1.05i)4-s + (−2.61 + 3.07i)5-s + (−2.40 − 1.27i)6-s + (−1.72 − 1.03i)7-s + (3.64 + 7.88i)8-s + (2.92 + 0.644i)9-s + (5.60 − 2.97i)10-s + (20.6 + 1.11i)11-s + (−1.71 − 2.02i)12-s + (1.53 + 6.97i)13-s + (1.91 + 2.51i)14-s + (−5.08 + 4.81i)15-s + (−0.407 − 7.51i)16-s + (6.88 − 4.14i)17-s + ⋯ |
| L(s) = 1 | + (−0.729 − 0.290i)2-s + (0.573 + 0.0624i)3-s + (−0.277 − 0.263i)4-s + (−0.523 + 0.615i)5-s + (−0.400 − 0.212i)6-s + (−0.246 − 0.148i)7-s + (0.456 + 0.985i)8-s + (0.325 + 0.0716i)9-s + (0.560 − 0.297i)10-s + (1.87 + 0.101i)11-s + (−0.143 − 0.168i)12-s + (0.118 + 0.536i)13-s + (0.136 + 0.179i)14-s + (−0.338 + 0.320i)15-s + (−0.0254 − 0.469i)16-s + (0.405 − 0.243i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.07862 + 0.128459i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07862 + 0.128459i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.72 - 0.187i)T \) |
| 59 | \( 1 + (-24.8 - 53.4i)T \) |
| good | 2 | \( 1 + (1.45 + 0.581i)T + (2.90 + 2.75i)T^{2} \) |
| 5 | \( 1 + (2.61 - 3.07i)T + (-4.04 - 24.6i)T^{2} \) |
| 7 | \( 1 + (1.72 + 1.03i)T + (22.9 + 43.2i)T^{2} \) |
| 11 | \( 1 + (-20.6 - 1.11i)T + (120. + 13.0i)T^{2} \) |
| 13 | \( 1 + (-1.53 - 6.97i)T + (-153. + 70.9i)T^{2} \) |
| 17 | \( 1 + (-6.88 + 4.14i)T + (135. - 255. i)T^{2} \) |
| 19 | \( 1 + (2.17 - 7.81i)T + (-309. - 186. i)T^{2} \) |
| 23 | \( 1 + (-23.1 + 15.6i)T + (195. - 491. i)T^{2} \) |
| 29 | \( 1 + (-15.4 - 38.7i)T + (-610. + 578. i)T^{2} \) |
| 31 | \( 1 + (-2.83 + 0.788i)T + (823. - 495. i)T^{2} \) |
| 37 | \( 1 + (-3.58 + 7.74i)T + (-886. - 1.04e3i)T^{2} \) |
| 41 | \( 1 + (13.3 - 19.6i)T + (-622. - 1.56e3i)T^{2} \) |
| 43 | \( 1 + (-19.6 + 1.06i)T + (1.83e3 - 199. i)T^{2} \) |
| 47 | \( 1 + (1.59 - 1.35i)T + (357. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + (33.9 - 64.1i)T + (-1.57e3 - 2.32e3i)T^{2} \) |
| 61 | \( 1 + (76.4 + 30.4i)T + (2.70e3 + 2.55e3i)T^{2} \) |
| 67 | \( 1 + (17.7 + 38.3i)T + (-2.90e3 + 3.42e3i)T^{2} \) |
| 71 | \( 1 + (63.6 + 74.9i)T + (-815. + 4.97e3i)T^{2} \) |
| 73 | \( 1 + (27.6 + 36.3i)T + (-1.42e3 + 5.13e3i)T^{2} \) |
| 79 | \( 1 + (-36.6 + 3.98i)T + (6.09e3 - 1.34e3i)T^{2} \) |
| 83 | \( 1 + (-16.4 - 48.7i)T + (-5.48e3 + 4.16e3i)T^{2} \) |
| 89 | \( 1 + (-35.0 + 13.9i)T + (5.75e3 - 5.44e3i)T^{2} \) |
| 97 | \( 1 + (-74.8 + 98.4i)T + (-2.51e3 - 9.06e3i)T^{2} \) |
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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
−12.27988690475176973952144024967, −11.28612489545358230957378068600, −10.40972250250091986919815444487, −9.289973814398654327583584022194, −8.815491301514542608176770953274, −7.46930120077410988652782473372, −6.45883205229757323443664197493, −4.56619712403154626775623605859, −3.31582992959642162246422781958, −1.39449733352929797205492208055,
0.985553867081686931083931026501, 3.46170602501190685784303768269, 4.47055439776498688126607846058, 6.41174358478179177531701558003, 7.54946072930997933998289161667, 8.515038372139573366644452198920, 9.111722314238181008254824523007, 9.972314666599654525138771894676, 11.56575974498726788087353137910, 12.47354854267956576520424553026
