| L(s) = 1 | + (2.07 − 0.329i)2-s + (−1.22 − 1.22i)3-s + (2.31 − 0.751i)4-s + (−2.11 + 0.740i)5-s + (−2.94 − 2.15i)6-s + (1.98 + 1.98i)7-s + (0.812 − 0.413i)8-s + (−0.0149 + 2.99i)9-s + (−4.14 + 2.23i)10-s + (−0.542 + 0.747i)11-s + (−3.74 − 1.92i)12-s + (0.794 − 5.01i)13-s + (4.77 + 3.46i)14-s + (3.48 + 1.68i)15-s + (−2.38 + 1.73i)16-s + (−2.38 − 4.67i)17-s + ⋯ |
| L(s) = 1 | + (1.47 − 0.232i)2-s + (−0.705 − 0.708i)3-s + (1.15 − 0.375i)4-s + (−0.943 + 0.330i)5-s + (−1.20 − 0.878i)6-s + (0.749 + 0.749i)7-s + (0.287 − 0.146i)8-s + (−0.00498 + 0.999i)9-s + (−1.31 + 0.706i)10-s + (−0.163 + 0.225i)11-s + (−1.08 − 0.554i)12-s + (0.220 − 1.39i)13-s + (1.27 + 0.927i)14-s + (0.900 + 0.435i)15-s + (−0.596 + 0.433i)16-s + (−0.577 − 1.13i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.34023 - 0.380393i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.34023 - 0.380393i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.22 + 1.22i)T \) |
| 5 | \( 1 + (2.11 - 0.740i)T \) |
| good | 2 | \( 1 + (-2.07 + 0.329i)T + (1.90 - 0.618i)T^{2} \) |
| 7 | \( 1 + (-1.98 - 1.98i)T + 7iT^{2} \) |
| 11 | \( 1 + (0.542 - 0.747i)T + (-3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.794 + 5.01i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (2.38 + 4.67i)T + (-9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (0.201 + 0.0655i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.0321 - 0.202i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (-0.160 - 0.494i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (2.03 - 6.27i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.33 - 0.370i)T + (35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (-4.36 - 6.00i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (0.516 - 0.516i)T - 43iT^{2} \) |
| 47 | \( 1 + (-8.19 - 4.17i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (-5.30 + 10.4i)T + (-31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (0.0415 - 0.0301i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (3.17 + 2.30i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (4.08 - 2.08i)T + (39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-3.13 + 1.01i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (11.2 - 1.78i)T + (69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (-3.90 + 1.27i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (13.9 - 7.11i)T + (48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (6.24 + 4.53i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (4.00 - 7.86i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−14.34878374463702860756610807505, −13.14195000319557849845693422104, −12.32827765737910324721644710737, −11.55336891103535964275064173796, −10.82662927360643027383591484392, −8.342095419307811531064188720156, −7.09443545133175768696595864996, −5.65227855766020109866271756451, −4.70302547271001067349902254523, −2.78089845567255212766530881673,
4.05244980253353912105105840716, 4.37287624686053762433316872017, 5.87135184515817219623244435392, 7.21827361417190819187189731878, 8.929408983589492290286605207188, 10.82140941108568671779959473259, 11.52444873247143160459253082508, 12.44017157227893458349569070282, 13.67010827503636388103946577634, 14.72995491605002404743146115443
