| 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 | \( 1 + (-0.527 + 3.33i)T + (-92.2 - 29.9i)T^{2} \) |
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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
−12.86483056118908527971198969749, −11.97782838696010143817251054667, −10.92417511132307293051062785670, −9.755519923195095329525257287568, −8.067797978637905491254319895333, −7.48469996552712524439236298651, −6.30366328508568003738979295598, −4.86100317551990405703741496984, −3.50560017481963541971941452183, −2.33512728686979501436632776764,
2.72049562003849283943005451411, 4.15362055146896126551383314604, 4.91877576455446823708507517277, 5.81985878242921899602193701882, 7.85239884505822452409782861243, 8.970222670478036467386435088947, 9.422911065173177088549765793229, 10.95988286654268609421846912043, 12.24048440223828099289135521290, 12.80322372699200652004422120814
