| L(s) = 1 | + (−0.437 − 0.0692i)2-s + (2.82 − 1.44i)3-s + (−1.71 − 0.557i)4-s + (−2.19 − 0.426i)5-s + (−1.33 + 0.434i)6-s + (−1.52 − 2.15i)7-s + (1.50 + 0.764i)8-s + (4.15 − 5.72i)9-s + (0.930 + 0.338i)10-s + (2.41 − 1.75i)11-s + (−5.65 + 0.895i)12-s + (0.482 + 3.04i)13-s + (0.518 + 1.05i)14-s + (−6.82 + 1.95i)15-s + (2.31 + 1.68i)16-s + (0.763 + 0.388i)17-s + ⋯ |
| L(s) = 1 | + (−0.309 − 0.0489i)2-s + (1.63 − 0.831i)3-s + (−0.857 − 0.278i)4-s + (−0.981 − 0.190i)5-s + (−0.545 + 0.177i)6-s + (−0.577 − 0.816i)7-s + (0.530 + 0.270i)8-s + (1.38 − 1.90i)9-s + (0.294 + 0.107i)10-s + (0.729 − 0.529i)11-s + (−1.63 + 0.258i)12-s + (0.133 + 0.844i)13-s + (0.138 + 0.280i)14-s + (−1.76 + 0.505i)15-s + (0.578 + 0.420i)16-s + (0.185 + 0.0942i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0435 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0435 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.848178 - 0.811988i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.848178 - 0.811988i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.19 + 0.426i)T \) |
| 7 | \( 1 + (1.52 + 2.15i)T \) |
| good | 2 | \( 1 + (0.437 + 0.0692i)T + (1.90 + 0.618i)T^{2} \) |
| 3 | \( 1 + (-2.82 + 1.44i)T + (1.76 - 2.42i)T^{2} \) |
| 11 | \( 1 + (-2.41 + 1.75i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.482 - 3.04i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-0.763 - 0.388i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-1.72 - 5.31i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.214 + 1.35i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (4.22 + 1.37i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-5.16 + 1.67i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.429 - 2.71i)T + (-35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (6.23 - 8.58i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-4.86 + 4.86i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.788 - 1.54i)T + (-27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (5.16 + 10.1i)T + (-31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-0.0470 - 0.0341i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.65 - 5.02i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (0.445 - 0.873i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (2.53 - 7.78i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (4.88 + 0.774i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-8.47 - 2.75i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (0.465 - 0.914i)T + (-48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (-8.07 + 5.86i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.309 - 0.607i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−12.79352056197147645376346277391, −11.68625567632260109677299631873, −10.05234528055535178979994711005, −9.206853829862697077028660513857, −8.358841498776739369013876060419, −7.68213664641176363352197713774, −6.55467565716017199460464839330, −4.15806722890216189835391751410, −3.48639218923762099961951934658, −1.22622687221267833104906664774,
2.95114316343993887646739582725, 3.79890568057485343726868627753, 4.90958055693443299871416182753, 7.26228002250110884109086419671, 8.146724411206944669426088130181, 9.029237694371464846459652612116, 9.472843323581224421440451440464, 10.60321637875156742421281960382, 12.16458472443954812157466830247, 13.10968054194848531699019188591
