| L(s) = 1 | + (0.321 − 2.03i)2-s + (1.39 + 2.73i)3-s + (−2.11 − 0.688i)4-s + (−0.390 − 2.20i)5-s + (5.99 − 1.94i)6-s + (1.56 + 2.12i)7-s + (−0.212 + 0.417i)8-s + (−3.76 + 5.18i)9-s + (−4.59 + 0.0857i)10-s + (2.92 − 2.12i)11-s + (−1.06 − 6.74i)12-s + (−2.07 + 0.329i)13-s + (4.83 − 2.50i)14-s + (5.47 − 4.13i)15-s + (−2.82 − 2.05i)16-s + (−0.704 + 1.38i)17-s + ⋯ |
| L(s) = 1 | + (0.227 − 1.43i)2-s + (0.803 + 1.57i)3-s + (−1.05 − 0.344i)4-s + (−0.174 − 0.984i)5-s + (2.44 − 0.795i)6-s + (0.593 + 0.804i)7-s + (−0.0752 + 0.147i)8-s + (−1.25 + 1.72i)9-s + (−1.45 + 0.0271i)10-s + (0.882 − 0.641i)11-s + (−0.308 − 1.94i)12-s + (−0.576 + 0.0913i)13-s + (1.29 − 0.668i)14-s + (1.41 − 1.06i)15-s + (−0.706 − 0.513i)16-s + (−0.170 + 0.335i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.716 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.48401 - 0.603634i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48401 - 0.603634i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.390 + 2.20i)T \) |
| 7 | \( 1 + (-1.56 - 2.12i)T \) |
| good | 2 | \( 1 + (-0.321 + 2.03i)T + (-1.90 - 0.618i)T^{2} \) |
| 3 | \( 1 + (-1.39 - 2.73i)T + (-1.76 + 2.42i)T^{2} \) |
| 11 | \( 1 + (-2.92 + 2.12i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (2.07 - 0.329i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (0.704 - 1.38i)T + (-9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (0.298 + 0.917i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (2.57 + 0.408i)T + (21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (4.65 + 1.51i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (9.20 - 2.99i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-9.27 + 1.46i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (3.01 - 4.15i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-0.604 - 0.604i)T + 43iT^{2} \) |
| 47 | \( 1 + (-7.74 + 3.94i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-6.14 + 3.12i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (0.161 + 0.117i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (3.90 + 5.37i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-1.34 - 0.687i)T + (39.3 + 54.2i)T^{2} \) |
| 71 | \( 1 + (-2.03 + 6.25i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.838 + 5.29i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (-15.4 - 5.03i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-5.09 - 2.59i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (5.62 - 4.08i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (2.15 - 1.09i)T + (57.0 - 78.4i)T^{2} \) |
| 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
−12.33128630552899325510576676843, −11.47523340448871133531621005660, −10.73599154109000432630696082757, −9.385218579667010816570649171823, −9.210305357062286778600313353371, −8.125733884426579698631404233041, −5.40871515039373544387130507959, −4.39921029954698125109429815729, −3.60495800479405572609126574992, −2.12860116518128533699323163755,
2.09191139427929287379854678734, 4.00780379608446943926426565027, 5.91700902733016295021302544639, 7.08998270063202666583923583231, 7.26871915095038527383573297499, 8.068937699402911499854326310077, 9.348448716312102959946007423741, 11.06168229192529509329618112708, 12.14067853336280972959720573966, 13.34218829624280653126030950741
