| L(s) = 1 | + (1.58 + 0.0830i)2-s + (1.23 + 1.52i)3-s + (0.513 + 0.0539i)4-s + (−0.221 − 2.22i)5-s + (1.82 + 2.51i)6-s + (1.63 + 2.08i)7-s + (−2.32 − 0.368i)8-s + (−0.174 + 0.822i)9-s + (−0.166 − 3.54i)10-s + (−2.03 + 0.433i)11-s + (0.550 + 0.848i)12-s + (−1.94 − 0.992i)13-s + (2.40 + 3.43i)14-s + (3.11 − 3.08i)15-s + (−4.66 − 0.990i)16-s + (1.81 + 4.72i)17-s + ⋯ |
| L(s) = 1 | + (1.12 + 0.0587i)2-s + (0.712 + 0.879i)3-s + (0.256 + 0.0269i)4-s + (−0.0992 − 0.995i)5-s + (0.746 + 1.02i)6-s + (0.616 + 0.787i)7-s + (−0.821 − 0.130i)8-s + (−0.0583 + 0.274i)9-s + (−0.0527 − 1.12i)10-s + (−0.614 + 0.130i)11-s + (0.159 + 0.244i)12-s + (−0.540 − 0.275i)13-s + (0.644 + 0.918i)14-s + (0.804 − 0.795i)15-s + (−1.16 − 0.247i)16-s + (0.440 + 1.14i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.06754 + 0.488191i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.06754 + 0.488191i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.221 + 2.22i)T \) |
| 7 | \( 1 + (-1.63 - 2.08i)T \) |
| good | 2 | \( 1 + (-1.58 - 0.0830i)T + (1.98 + 0.209i)T^{2} \) |
| 3 | \( 1 + (-1.23 - 1.52i)T + (-0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (2.03 - 0.433i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (1.94 + 0.992i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-1.81 - 4.72i)T + (-12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (0.785 + 7.47i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (0.105 - 2.00i)T + (-22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (4.12 - 5.67i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.58 + 5.80i)T + (-20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-2.22 + 1.44i)T + (15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (-8.47 + 2.75i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-0.986 - 0.986i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2.45 - 0.941i)T + (34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (7.17 - 5.81i)T + (11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (7.04 - 7.81i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (6.37 - 5.73i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-3.06 + 1.17i)T + (49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (-5.41 - 3.93i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.51 + 6.94i)T + (-29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (-1.08 - 2.44i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-0.456 + 2.88i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-8.47 - 9.41i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (-0.527 - 3.33i)T + (-92.2 + 29.9i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.80322372699200652004422120814, −12.24048440223828099289135521290, −10.95988286654268609421846912043, −9.422911065173177088549765793229, −8.970222670478036467386435088947, −7.85239884505822452409782861243, −5.81985878242921899602193701882, −4.91877576455446823708507517277, −4.15362055146896126551383314604, −2.72049562003849283943005451411,
2.33512728686979501436632776764, 3.50560017481963541971941452183, 4.86100317551990405703741496984, 6.30366328508568003738979295598, 7.48469996552712524439236298651, 8.067797978637905491254319895333, 9.755519923195095329525257287568, 10.92417511132307293051062785670, 11.97782838696010143817251054667, 12.86483056118908527971198969749
